QC 


QUESTIONS  AND  EXEECMS 


STEWART'S  LESSONS 


ELEMENTARY    PHYSICS. 


-NRLF 


110 


BOS  TO 

A  X  1)      1!  K  A  T  II 


GIFT  OF 
ENGINEERING  LIBRARY 


QUESTIONS  AND  EXERCISES 


ON 


STEWART'S  LESSONS 


IN 


ELEMENTAKY   PHYSICS. 


GEOEGE   A.  HILL, 
4 

ASSISTANT  PROFESSOR   OF  PHYSICS   IN   HARVARD   UNIVERSITY. 


WITH  ANSWERS  AND  OCCASIONAL  SOLUTIONS. 


BOSTON: 

GINN     AND     HEATH. 

1880. 


Entered  according  to  Act  of  Congress,  in  the  year  1874, 

BY    GINN    BROTHERS, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


GIFT  OF 
ENGINEERING  LIBRARY 


PRESS  OF  ROCKWELL  AND  CHURCHILL, 
39  Arch  St.,  Boston. 


t 

PREFACE. 


THE  following  pages  have  been  drawn  up  with  the  aim  of 
making  Mr.  Stewart's  excellent  work  more  useful  in  element- 
ary teaching. 

Part  I.  consists  of  questions  upon  the  text  of  Mr.  Stewart's 
book  which  are  intended  to  be  direct  and  exhaustive.  Opin- 
ions differ  as  to  the  value  of  such  questions.  No  doubt  a 
thoroughly  competent  teacher  will  ask  questions  in  his  own 
way  with  the  best  effect ;  but  unfortunately  such  teachers,  at 
least  in  scientific  subjects,  are  not  numerous.  In  all  cases  the 
questions  will  be  found  useful  for  review  and  examination 
purposes. 

Parts  II.  and  III.,  which  form  the  principal  part  of  the 
work,  have  been  written  with  two  objects  in  view.  First,  to 
stimulate  original  thought  on  the  part  of  the  student,  and  to 
give  the  teacher  the  means  of  testing  thoroughly  the  student's 
knowledge  of  principles.  Secondly,  to  make  certain  needful 
additions  to  the  felicitous  but  cursory  sketch  of  Mechanics, 
Hydrostatics,  and  Pneumatics,  contained  in  the  first  two 
chapters  of  Mr.  Stewart's  book. 

Molecular  Physics  is  rapidly  assuming  the  character  of  an 
exact  science  ;  and  in  proportion  as  this  takes  place,  the  im- 
portance of  a  good  knowledge  of  the  general  laws  of  Motion 
and  Force,  and  of  the  ability  to  reason  deductively,  increases. 
Nothing  can  give  training  in  deduction  better  than  the  study 
of  Rational  Mechanics.  Training  in  the  methods  of  induc- 
tion, which  is  so  large  a  part  of  scientific  culture,  cannot,  in 

865704 


iv  PREFACE. 

our  judgment,  be  imparted  successfully  by  the  study  of  text- 
books ;  the  place  to  receive  it  is  in  Physical  Laboratories, 
which  are  happily  becoming  more  and  more  common,  or  by 
observation  and  reflection  in  the  vast  Laboratory  of  Nature 
around  us.  The  chief  value,  which  the  text-book  study  of 
Physics  can  be  made  to  have,  consists  in  disciplining  the 
mind  in  scientific  demonstration  of  the  deductive  kind. 

The  Exercises  are  divided  into  two  classes,  as  explained  on 
page  69.  A  few  of  them  are  original ;  the  most  have  been 
selected  from  English  works.  Few  of  them  require  much 
numerical  work,  and  many  of  them  none  at  all. 

In  preparing  the  Solutions,  the  author  has  been  under 
obligations  to  the  elementary  writings  of  Professors  Thomson, 
Tait,  and  Maxwell. 

It  was  found  impossible  to  prepare  solutions  of  the  more 
difficult  Exercises  in  small  type  in  season  for  the  present 
edition.  In  working  out  these  Exercises,  where  aid  is  found 
necessary  it  should  be  obtained,  if  possible,  from  a  competent 
teacher.  The  student  is  also  strongly  advised  to  consult 
special  works  which  treat  of  the  subjects  covered  by  the  Ex- 
ercises. By  doing  this,  the  student  will  not  merely  find  the 
aid  which  he  desires,  —  he  will  be  acquiring  a  habit  of  mind 
which  is  characteristic  of  the  cultivated  man  and  of  all  pro- 
ductive scholarship,  the  habit  of  consulting  and  carefully 
comparing  the  views  which  different  minds  take  of  the  same 
subject,  and  of  that  originality  in  thought  which  comes  from 
an  independent  use  of  many  authorities.  On  page  69  will  be 
found  a  list  of  elementary  works  which  may  be  consulted  with 
advantage. 

G.   A.   HILL. 

CAMBRIDGE,  August  25, 1874. 


CONTENTS. 


PAGE 
PART  I.    QUESTIONS 1-68 

II.    EXERCISES  AND  PROBLEMS         .        .       69-114 

III.     ANSWERS  AND  SOLUTIONS       .        .         115-168 
Answers  .......     115 

Solutions 127 

APPENDIX 169-188 

I.     English  Weights  and  Measures .         .        .         169 
II.     United  States  Weights  and  Measures     .         .     173 

III.  Metric  Weights  and  Measures   .        .        .         175 

IV.  Mathematical  Formulae         .        .        .        .179 
V.     Physical  Tables 185 


PART    I. 

QUESTIONS  ON  STEWAET'S  ELEIENTAEY 
PHYSICS. 


Introduction. 

1.  How  do  we  become  aware  of  the  existence  of  objects 
outside  of  ourselves  ? 

2.  What   is  the  ground  of  our  expectation  that  the 
sun  will  rise  to-morrow  ?     In  general,  when  is  our  expec- 
tation that  a  certain  phenomenon  will  recur  well  grounded  1 

3.  What  are  characteristics  of  the  knowledge  of  physi- 
cal laws  which  men  acquire  from  every-day  experience  ? 

4.  How  long  since  men  first  set  themselves  systemati- 
cally to  the  task  of  acquiring  a  knowledge  of  the  laws  of 
nature  ? 

5.  What  is  the  object  of  Physics  ? 

6.  What  do  we  learn  from  astronomy  concerning  the 
magnitude  of  the  Universe  ? 

7.  Explain  the  three-fold  division  of  matter  into  sub- 
stances, molecules,   and  atoms.     Illustrate   by  a  familiar 
example. 

8.  What  is  the  analogous  three-fold  division  in  astron- 
omy ? 

9.  What  resemblance  exists  between  the  structure  of 
the  Universe  and  that  of  a  body  on  the  earth's  surface, 
in  consequence  of  which  both  may  be  called  porous  ? 

10.  Distinguish  between  physical  and  sensible  pores. 
What  proves  the  existence  of  physical  pores  ?     Give  in- 
stances of  bodies  having  sensible  pores. 

1  A 


2  QUESTIONS  ON  STEWART'S      [INTROD. 

11.  Name  the  three  states  of  matter.     What  are  their 
chief  characteristics  ?     Give  examples  of  each  state. 

12.  Explain  relative  motion  by  means  of  the  motions- 
of  the  earth  and  the  planets. 

13.  What  have  we  strong  reason  to  suppose  to  be  the 
condition  of  any  substance  at  rest,  —  the  molecules  of  a 
block  of  stone,  for  example  1 

14.  In  view  of  our  present  scientific  knowledge,  what 
may  be  asset- te.d  of -every  body  in  the  universe,  lame  or 
snvill  ? 

15.  Illustrate  by  examples  the  meaning  of  the  term 
;  forve. 

16.  Mention  three  of  the  most  universal  forces  of  na- 
ture.'   What  are  their  effects  ?     What  consequences  would 
ensue  if  these  forces  severally  were  to  cease  to  exist  ? 

17.  What  must  be  the  effect  of  a  single  force  acting 
upon  a  body  ?     What  may  be  the  effect  of  two  or  more 
forces  acting  simultaneously  on  a  body  1     Illustrate  by 
the  force  of  gravitation. 


LESS.  L]  ELEMENTARY  PHYSICS. 


CHAPTER  I. 

LAWS   OF  MOTION. 

LESSON  I.  —  Determination  of  Units. 

18.  What  is  the  unit  of  time  1 

19.  Mention  the  two  chief  advantages  of  the  Metric 
system  of  weights  and  measures. 

20.  What  is  the  unit  of  length  in  the  Metric  system, 
and  its  value  approximately  in  English  inches  ? 

21.  Enumerate  the  chief  multiples  and  sub-multiples 
of  the  metre,  and  give  their  values  in  terms  of  the  metre. 

22.  How  are  units  of  surface  and  of  capacity  derived 
from  those  of  length  ?     Give  examples  of  each  kind. 

23.  What  is  the  value  in  square  metres  of  an  are  ?  of 
a  centiare  ?  of  a  hectare  ? 

24.  What  is  a  litre  ?    Name  some  of  its  decimal  multi- 
ples and  sub-multiples. 

25.  What  is  the  ratio  between  two  successive  units  of 
length,  as  the  centimetre  and  the  decimetre  ?    between 
the  two  corresponding  units  of  surface  ?  between  the  two 
corresponding  units  of  volume  ? 

26.  How  many   square  feet  are  there  in  150  square 
inches  ?     How  many  square  centimetres  are  there  in  150 
square  millimetres  ? 

27.  How  many  cubic  yards  are  there  in  93  cubic  feet  ? 
How  many  litres  are  there  in  1789  millilitres  1 

28.  What  superiority  of  the  Metric  system  is  established 
by  such  questions  as  26  and  27  ? 

29.  What  is  the  unit  of  mass  in  the  Metric  system,  and 
how  is  it  connected  with  the  unit  of  length  1 

30.  Enumerate  the  chief  derivative  units  of  mass,  and 
give  their  values  in  terms  of  the  gramme. 


4  QUESTIONS  ON  STEWART'S      [CHAP.  i. 

31.  Illustrate  the  meaning  of  velocity  by  the  example 
of  a  railway  train.     How  would  you  define  the  word  ? 

32.  Show  that  the  space  passed  over  by  a  body  moving 
for  any  time  with  a  uniform  velocity  is  equal  to  the  velo- 
city multiplied  by  the  time. 

33.  What  is  a  convenient  unit  of  velocity  ? 

34.  How  may  the  relative  masses  of  bodies  of  the  same 
kind  be  estimated  1 

35.  Why  cannot  weight  be  adopted  as  a  fundamental 
method  of  measuring  mass  ? 

36.  What  is  the  ultimate  test  that  two  different  sub- 
stances have  the  same  mass  1 

37.  What  relation  between  mass  and  weight  has  been 
established  which  enables  us  to  employ  weight  as  a  con- 
venient practical  means  of  estimating  mass  ? 

38.  Define  the  unit  of  force. 

39.  It  is  true  that  it  requires  a  double  force  to  produce 
either  (1)  the  same  velocity  in  a  double  mass,  or  (2)  a 
double  velocity  in  the  same  mass  ;  show  that  one  of  these 
truths  follows   immediately  from  the  definition  of   the 
unit  of  force? 


LESSON  II.  —  First  Law  of  Motion. 

40.  What  does  the  first  law  of  motion  assert  1 

41.  Explain  how  this  law  is  apparently,  but  not  really, 
contradicted  by  every-day  experience. 

42.  What  are  the  two  great  forces  which  tend  to  stop  all 
motion  on  the  surface  of  the  earth  ?    Give  illustrations  of 
each. 

43.  What  is  the  nearest  approach  to  a  perpetual  motion 
with  which  we  are  acquainted  ? 

44.  Explain  the  following  illustrations  of  the  first  law 
of  motion  : — 

1.  A  man  is  on  horseback,  and  the  horse  starts  off 
suddenly.     In  what  direction  will  the  man  fall  ? 

2.  A  man  is  on  horseback,  and  the  horse  stops  sud- 
denly.    In  what  direction  will  the  man  fall  1 


LESS,  in.]         ELEMENTARY  PHYSIOS.  5 

45.    Show  how  the  first  law  of  motion  serves  to  explain 
some  of  the  common  phenomena  of  rotation. 


LESSON  III.  —  Second  Law  of  Motion. 

46.  State  the  second  law  of  motion. 

47.  For  the  sake  of  clearness,  what  two  cases  may  be 
considered  separately  under  this  law  1 

48.  Suppose  that  a  ball  is  thrown  upwards  or  sideways 
in  a  moving  railway-carriage  ;  show  that  its  motion  rela- 
tive to  the  carriage  is  different  from  its  motion  relative  to 
the  ground,  and  that  the  motion  relative  to  the  ground  is 
represented  by  the  diagonal  of  a  parallelogram,  the  sides  of 
which  represent  the  motions  of  the  ball  and  of  the  car- 
riage respectively. 

49.  If  I  leap  vertically   upwards   at   the   equator,  I 
alight  upon  the  place  from  which  I  sprang,  although  all 
places  on  the  equator  are  moving,  in  consequence  of  the 
earth's  rotation,  at  the  rate  of  about  one  mile  in  three  sec- 
onds ;  explain  this  by  means  of  the  first  and  second  laws 
of  motion. 

50.  A  balloon  at  the  height  of  two   miles  above  the 
earth's  surface  is  totally  immersed  in,  and  carried  along 
with,  a  current  of  air  moving  at  the  rate  of  60  miles  an 
hour.     A  feather  is  dropped  over  the  edge  of  the  car  ; 
will  it  be  blown  away,  or  will  it  appear  to  drop  vertically 
down  1 

51.  A  ship  is  in  rapid  motion,  and  a  stone  is  dropped 
from  the  top  of  the  mast  ;  where  will  it  fall  ? 

52.  Examine  the  case  in  which  a  force  produces  motion 
in  the  same  direction  as  an  already  existing  motion,  as 
when  a  ball  is  thrown  directly  forwards  in  a  moving  rail- 
way-carriage. 

53.  Discuss  the  following  example  of  motion  in  a  ver- 
tical direction :  — 

A  movable    chamber   4*9  m.  high  can  be  made,  by 
machinery,  to  descend  the  vertical  shaft  of  a  mine  with 


6  QUESTIONS  ON  STEWARTS      [CHAP.  I. 

the  uniform  velocity  of  9'8  m.  per  second.  A  ball 
is  dropped  from  the  top  of  the  chamber,  (1)  when  the 
chamber  is  at  rest,  (2)  when  the  chamber  is  descending 
with  the  uniform  velocity  of  9*8  m.  per  second. 

54.  If  a  stone  be  dropped  from  the  top  of  a  cliff,  what 
velocity  will  it  acquire  under  the  action  of  gravity  in  one 
second  1  in  two  seconds  ?  in  t  seconds  ?  in  one  quarter  of 
a  second  ? 

55.  Explain  with  precision  the  statement  that  "  at  the 
end  of  one  second  a  body  falling  freely  will  attain  a  ve- 
locity of  9*8  m.  per  second/' 

56.  What  may  be  called  the  average  or  mean  velocity 
of  a  falling  body  during  the  first  second  ?  during  the  first 
two  seconds  1  during  t  seconds  ? 

57.  Prove  that,  in  uniform  motion,  space  passed  over 
is  equal  to  velocity  multiplied  by  time. 

58.  Show  that  any  case  of  uniform  motion  may  be  rep- 
resented graphically  by  the  area  of  a  rectangle. 

59.  Prove,  by  dividing  a  second  into  tenths,  and  sup- 
posing the  motion  uniform  during  each  tenth  of  the  second, 
that  the  space  passed  over  by  a  body  falling  freely  in  the 
first  second  of  its  motion  is  4'9  metres. 

60.  In  general,  what  represents  the  space  described  by 
a  body  falling  freely  for  any  given  time  ? 

61.  If  t  =  the  whole  time  of  fall,  and  s  =  the  space 
passed  over,  show,  from  what  has  already  been  established, 
that  s  =  4-9  tz. 

62.  Comparing  the  results  in  questions  56  and  59  it 
appears  that  the  space  passed  over  in  the  first  second  of 
motion  is  equal  numerically  to  the  mean  velocity  during 
that  second.     Accepting  this  relation  as  true  in  general 
(which  is  the  fact),  find  the  space  described  during  the 
second  second  of  motion  ;  during  the  third  second. 

63.  Suppose  a  projectile,  as  a  bomb-shell,  to  be  fired 
obliquely  into  the  air ;  prove  that  its  actual  path  under 
the  action  of  gravity  will  be  a  curve,  bending  farther  and 
farther  from  the  original  line  of  impulse.     What  may 
this  curve  be  shown  to  be  ? 


LESS,  iv.]          ELEMENTARY  PHYSICS.  7 

LESSON  IV.  —  Second  Law  of  Motion  (continued). 

64.  Suppose  a  piece  of  iron  to  fall  by  the  action  of 
gravity,  and  also  to  be  acted  upon  by  a  magnet  so  placed 
as  to  give  it  in  one  second  a  velocity  in  the  same  direc- 
tion as  gravity  of  9*8  m.  per  second  ;  find  the  velocity 
acquired  and  the  space  described  in  one  second. 

65.  From  the  results  in  question  64,  what  may  be  in- 
ferred to  be  the  proper  measure  of  different  forces  applied 
to  the  same  body  ? 

66.  What  have  we  found  to  be  the  measure  of  forces 
which  generate  the  same  velocity  in  bodies  having  differ- 
ent masses  ? 

67.  In  general,  what  product  represents  the  magnitude 
of,  or  is  the  measure  of,  a  force  ? 

68.  What  product  measures  the  momentum  of  a  moving 
body  ?     Define  the  measure  of  a  force  in  terms  of  the  mo- 
mentum which  it  will  generate. 

69.  Give  an  instance  of  two  forces  acting  in  different 
directions  simultaneously  on  a  body  at  the  same  point, 
and  determine  the  path  which  the  body  will  take  under 
the  joint  action  of  the  two  forces. 

70.  Explain  how  a  straight  line  may  be  employed  to 
represent  the  point  of  application,  the  direction,  and  the 
magnitude  of  a  force. 

71.  What  are  the  two  chief  results  respecting  the  sec- 
ond law  of  motion  reached,   one  in  the  last  Lesson,  the 
other  in  the  present  Lesson  ? 

72.  Give  examples  of  forces  which  act  in  such  a  way 
as  to  compel  a  body  to  remain  at  rest. 

73.  What  really  happens  when  a  heavy  body  rests  on 
the  floor  ? 

74.  What  is  the  effect  of  a  force  which  acts  on  a  body 
without  changing  its  state  of  rest  or  motion  called  ? 

75.  When    must   two  pressures,  or  statical  forces,  be 
considered  equal  to  each  other  ? 

76.  A  man  in  a  carriage  supports  a  half-hundred  weight 
in  his  hand.     The  carriage  and  all  that  it  contains  is  now 


8  QUESTIONS  ON  STEWARTS       [CHAP.  i. 

in  the  act  of  falling  over  a  precipice.     Will  lie  still  con- 
tinue to  feel  the  strain  of  the  weight  upon  his  arm  ? 

77.  A  weight  equal  to  100  kilogrammes  rests  upon  a 
support,  the  weight  of  which  support  may  be  neglected. 
This  support  is  not  altogether  prevented  from  falling,  but, 
in  virtue  of  the  machinery  with  which  it  is  connected, 
it  is  only  allowed  to  acquire  a  velocity  of  4*9  metres  in 
one  second.     What  will  be  the  pressure  on  the  support  1 

78.  What  two  ways   are  there  of  viewing  the  action 
of  two  forces  acting  simultaneously  on  a  body  in  different 
directions  ? 


,  LESSON  V.  —  Forces  Statically  Considered. 

79.  What  is  the  "  parallelogram  of  forces  "  1 

80.  Give  a  definition  of  the  resultant  of  two  forces 
acting  along  lines  which  intersect  one  another. 

81.  Distinguish  between  the  resultant  of  two  forces 
and  what  may  be  called  their  balancing  force. 

82.  If  two  forces,  P  =  6,  Q  =  8,  act  on  a  point  at  right 
angles  to  each  other,  find  the  numerical  value  of  their  re- 
sultant. 

83.  Describe  an  experimental  method  of  demonstrating 
the  truth  of  the  parallelogram  of  forces. 

84.  Suppose  that  we  have  two  parallel  forces  in  the  form 
of  weights  acting  at  the  ends  of  a  straight  horizontal  rigid 
bar  or  lever,  the  whole  resting  on  a  fixed  point  or  fulcrum 
between  the  forces.     Neglecting  the  weight  of  the  bar, 
under  what  condition  will  the  system  be  in  equilibrium  ? 
What  will  be  the  pressure  on  the  fulcrum  1 

85.  Define  the  moment  of  a  force  with  respect  to   a 
point. 

86.  State  the  condition  of  equilibrium  of  a  lever  in  the 
language  of  moments. 

87.  Give   examples  of  levers  where  a  comparatively 
small  force  at  a  great  leverage  produces  a  very  great  effect. 
What  do  you  mean  by  a  "  great  leverage  "  ? 


LESS,  vi.]          ELEMENTARY  PHYSICS.  9 

88.  On  a  lever,  the  perpendicular  distances  of  the  lines 
of  action  of  the  forces  from  the  fulcrum  are  often  called 
the  arms  of  the  lever.     What  relation  must  exist  between 
the  lengths  of  the  arms  in  order  that  a  given  force  applied 
at  the  end  of  one  arm  may  overcome  a  greater  force  ap- 
plied at  the  end  of  the  other  arm  ? 

89.  What  is  the  condition  of  equilibrium  of  any  num- 
ber of  forces  applied  in  one  plane  to  a  body  which  is  sup- 
ported on  a  fixed  point  or  fulcrum  ? 

90.  On  a  straight  lever  without  weight  we  have  on  the 
right  hand  of  the  fulcrum  two  forces,  namely,  8  grammes 
at  a  distance  of  6  centimetres,  and  12  grammes  at  a  dis- 
tance of  8  centimetres  ;  while  on  the  left  hand  we  have 
10  grammes  at  a  distance  of  10  centimetres.     Which  arm 
will  tend  to  fall  ? 


LESSON  VI.  —  Third  Law  of  Motion. 

91.  In  the  third  law  of  motion  what  is  asserted  of  any 
force  which  alters  the  state  of  rest  or  motion  of  a  body  as 
a  whole  ?     Give  an  illustration. 

92.  What  does  the  third  law  of  motion  assert  of  the 
momenta  generated  in  the  parts  of  a  body  or  system  of 
bodies  by  the  action  of  internal  forces  1     Illustrate  this 
truth  by  the  example  of  firing  a  gun. 

93.  How  is  the  third  law  of  motion  sometimes  stated  ? 

94.  Illustrate  this  law  by  the  example  of  a  stone  fall- 
ing to  the  ground. 

95.  How  does  the  discharge  of  a  cannon  which  is  firmly 
fixed  to  the  ground  furnish  another  illustration  of  the 
same  law  ? 

96.  According  to  this  law  of  motion,  what  must  take 
place  whenever  a  man  leaps  upward  from  the  ground  ? 

97.  Suppose  a  bomb-shell  flying  along  with  a  velocity 
of  200  m.  per  second  explodes  into  two  parts  of  equal 
weight,  one  of  which  is  propelled  forwards  in  the  exact 
direction  in  which  the  shell  is  moving  with  an  additional 

1* 


10  QUESTIONS  ON  STEWART'S       [CHAP.  i. 

velocity  of  200  m.  per  second.  Show,  by  means  of  the 
third  law  of  motion,  that  the  other  half  of  the  shell  will 
be  brought  to  rest  in  consequence  of  the  explosion. 

98.  Explain  the  Eolipyle. 

99.  Explain  the  ascent  of  a  rocket. 


LESS,  vii.]         ELEMENTARY  PHYSICS.  11 


CHAPTER  II. 

THE  FORCES   OF  NATURE. 

LESSON  VII.  —  Universal  Gravitation. 

100.  Into  what  three  groups  may  the  forces  of  nature 
be  divided  ? 

101.  What  is  the  distinction  between  molecular  and 
atomic  forces  ] 

102.  Illustrate  the  general  fact  that  some  of  the  forces 
connected  with  molecules  and  atoms  may  be  characterized 
as  permanent  while  others  are  temporary  and  evanescent. 

103.  What  is  the  most  important  and  best  understood 
force  belonging  to  matter  ] 

104.  What  question  respecting  terrestrial  gravity  did 
Newton  ask  himself,  and  what  answer  did  he  find  by  ex- 
periment ? 

105.  What  opinion  on  this  subject  was  held  by  the  fol- 
lowers of  Aristotle  ? 

106.  How  did  Galileo  overturn  the  Aristotelian  dogma  ? 

107.  Describe  the  "  guinea  and  feather "  experiment, 
stating  clearly  what  it  proves. 

108.  What  prevents  us  from  making  exact  experiments 
on  bodies  falling  freely  ? 

109.  What  effect  would  changes  in  the  force  of  gravity 
have  on  the  oscillations  of  a  pendulum  1 

110.  Describe  Newton's  pendulum  experiments,   and 
show  that  they  prove  that  the  weight  of  a  body  is  directly 
proportional  to  its  mass. 

111.  Compare  gravity  with  magnetism,  as  regards  the 
relation  between  the  acting  force  and  the  mass  acted  upon. 

112.  Show  that  the  measure  of  the  force  of  gravity 
which  acts  on  one  gramme  is  equal  to  9*8.     What  will  it 
be  on  5  grammes  1 


12  QUESTIONS  ON  STEWART'S      [CHAP.  n. 

113.  How  is  the,  vertical  direction  defined  ?     How  found 
by  experiment  ? 

114.  Why  are  plumb  lines  not  strictly  parallel  ?     What 
change  in  the  direction  of  a  plumb  line  is  produced  by 
travelling  one  mile  on  the  earth's  surface  1 

115.  What  effect  on  the  weight  of  a  body  would  be 
produced  by  a  change  in  the  mass  of  the  earth  or  attract- 
ing body  ? 

116.  State  the  law  of  "  inverse  squares,"  or  law  which 
expresses  the  mathematical  relation  between  the  distance 
of  two  bodies  from  each  other  and  the  force  of  attraction 
between  them. 

117.  Prove  the  law  of  "  inverse  squares  "  by  the  New- 
tonian method  of  comparing  the  force  of  the  earth's  attrac- 
tion at  the  moon  with  the  same  at  the  earth's  surface. 

118.  Give  the  complete  statement  of  the  law  of  univer- 
sal gravitation. 

119.  Illustrate  this  law  by  supposing  different  numer- 
ical values  for  the  attracting  masses  and  their  distance 
from  each  other. 


LESSON  VIII.  —  Atwood's  Machine. 

120.  What    is     the    object     of   Atwood's    Machine  ? 
Describe  its  chief  parts. 

121.  Describe  Experiment  A,  and  state  what  it  proves. 

122.  Describe  Experiment  B.  Compare  the  results  in  ex- 
periments A  and  B,  and  state  the  law  which  they  establish. 

123.  Describe  Experiment  C.     What  may  be  concluded, 
(1)  by  comparing    together    experiments   A   and    C,  (2) 
by  comparing  together  the  results  in  all  three  experiments, 
A,  B,  and  C  ? 

124.  Describe  Experiment  D. 

125.  Describe  Experiment  E.     What  truth  do  experi- 
ments D  and  E  illustrate  ? 

126.  Describe  Experiment  F.    What  is  "  the  law  of  ve- 
locities "  which  is  demonstrated  by  this  experiment  ? 


LESS,  ix.]          ELEMENTARY  PHYSICS.  13 

127.  Describe  Experiment   G,  and  give   the  '-'law  of 
spaces  "  which  it  proves. 

128.  Show  that,  if  a  body  be  projected  vertically  up- 
wards, the  height  attained  is  proportional  to  the  square 
of  the  velocity  of  projection. 

129.  What  is  the  relation  in  theory  between  the  velo- 
city of  projection  of  a  stone,  and  the  velocity  with  which 
it  strikes  the  ground  on  its  return  ?     How  is  this  truth 
illustrated  by  Experiment  H 1 

130.  Neglecting  the  weight  of  the  pulley  in  Atwood's 
Machine,  let  the  one  box  weigh  600  grammes,  and  the 
other  400  ;  what  will  be  the  tension  of  the  string  during 
the  downward  motion  of  the  heavier  box  1 

131.  A  body  is  projected  vertically  upwards   with   a 
velocity  equal  to  19 '6  metres  per  second  ;  what  will  be  its 
velocity  after  it  has  risen  14'7  metres  ? 

132."  Give  a  brief  recapitulation  of  the  facts  connected 
with  the  action  of  gravity  at  the  earth's  surface. 


LESSON  IX.  —  Centre  of  Gravity,  etc. 

133.  Show  how   the  force  which  gravity  exerts  on  a 
body  may  be  resolved  into  a  system  of  parallel  forces,  and 
from  this  point  of  view  give  a  definition  of  the  centre  of 
gravity  of  a  body.- 

134.  Describe  a  simple  practical  way  of  finding  the  cen- 
tre of  gravity  of  a  body. 

135.  If  we  have  a  heavy  solid  resting  on  a  base,  what 
condition  must  be  fulfilled  in  order  that  it  may  remain  at 
rest  ?     Prove  the  necessity  of  this  condition. 

136.  Define  stable  equilibrium  and  unstable  equilibrium, 
and  give  examples  of  each. 

137.  State  a  simple  law   which  will  always    decide 
whether   an   equilibrium   is   stable   or   unstable.     What 
grounds  are  given  for  the  truth  of  this  law  ?     Illustrate 
its  application  by  the  example  of  the  egg. 

138.  Define  neutral  equilibrium,  arid  give  an  example. 


14  QUESTIONS  ON  STEWART'S     [CHAP.  n. 

139.  A  cone  is  placed  on  its  apex  on  a  flat  horizontal 
surface  ;  determine  the  kind  of  equilibrium. 

140.  A  uniformly  heavy  circular  wooden  disk  has  a 
piece  of  its  substance  taken  out,  and  a  piece  of  lead  insert- 
ed instead.     In.  what  position  will  it  rest  on  a  flat  hori- 
zontal surface  ? 

141.  How  will  a  man  rising  in  a  boat  affect  its  stability  ? 

142.  Why  is  a  cart  loaded  with  hay  more  liable  to  be 
overturned    from    irregularities    in  the  road  than     one 
loaded  with  the  same  weight  of  lead  ? 

143.  Describe  briefly  the  balance,  and  show  that  a  sen- 
sitive balance  enables  us  to  ascertain  with  great  exactness 
the  weight  of  a  body. 

144.  Can  you  determine  what  must  be  the  position  of 
the  centre  of  gravity  of  a  balance  relatively  to  the  centre  of 
suspension  in  order  that  the  balance  may  be  very  delicate  ? 

145.  Explain  the  use  of  the  pendulum,  (1)  in  detect- 
ing changes   in  the   force  of  gravity,  (2)  in  regulating 
clocks. 

146.  What  is  meant  by  the  isochronism  of  a  pendulum, 
and  how  was  it  first  discovered  ?     What  is  the  length  of  a 
seconds  pendulum  ? 

147.  What  is  the  law  which  expresses  the  relation  be- 
tween the  time  of  oscillation  of  a  pendulum  and  its 
length  1  

LESSON  X.  —  Forces  exhibited  in  Solids. 

148.  Describe  briefly,  and  illustrate,  the  chief  attractive 
forces  which  are  exhibited  in  bodies. 

149.  Describe  briefly,  and  exemplify,  the  following  re- 
sistances  to  deformation  which  are  called  into  action  by 
Various  forces  tending  to  alter  the  shape  of  a  solid  body  : 

(1)  Resistance  to  linear  extension. 

(2)  Resistance  to  linear  compression. 

(3)  Resistance  to  cubical  compression. 

(4)  Resistance  to  torsion. 

(5)  Resistance  to  flexure. 


LESS,  x.]  ELEMENTARY  PHYSIOS.  15 

Which  one  of  these  resistances  is  also   exhibited  by 
liquids  and  gases  ? 

150.  Explain  what  friction  is,  and  define  the  coefficient 
of  friction. 

151.  Prove  that  if  the  pressure  remain  the  same,  the 
friction  is  independent  of  the  magnitude  of  the  surface. 

152.  Give  Rennie's  laws  upon  friction.    How  may  fric- 
tion be  reduced  to  a  minimum  1 

153.  What  conditions  are  favorable  to  the  formation  of 
crystals  ?     Give  an  instance. 

154.  Give  instances  of  crystals  of  great  value  which 
have  not  yet  been  formed  artificially. 

155.  What  conclusion  about  crystals  is  drawn  from  the 
behavior  of  many  crystals,  —  for  example,  those  of  Ice- 
land spar  ? 

156.  What  peculiarities  of  structure  and  what  conse- 
quent properties  are  exemplified  by  such  substances  as 
wood  and  flax  ?  by  wrought-iron  ?  by  such  substances  as 
mica  and  oyster-shells  ? 

157.  Give  examples  of  solids  exhibiting  no  apparent 
trace  of  structure.     In  general,  what  effect  does  time  and 
vibration  have  on  the   structure   of  a   solid1?     Give  an 
example. 

158.  Define  tenacity.     How  are  experiments  on  tenaci- 
ty conducted  1    What  law  is  established  by  the  experi- 
ments ?    What  convenient  measure  of  tenacity  does  this 
law  point  out  ? 

159.  What  do  the  results  of  Wertheim's  experiments, 
as  given  in  the   table,  show  as  to  the  effect  of  time  on 
tenacity  ? 

160.  What   peculiarity  is   there  in   the   tenacity   of 
fibrous  solids  like  wood  ?     According  to  Musschenbroeck, 
which  is  the  most  tenacious  of  woods  1 

161.  Instead  of  bodies   suddenly  giving  way  under 
forces  tending  to  pull  their  particles  asunder,  what  other 
form  of  rupture  is  common  ? 

162.  What  does  ductility  denote  ?    Give  examples  of 
bodies  which  possess  it  ? 


16  QUESTIONS  ON  STEWART'S     [CHAP.  n. 

163.  What  is  malleability  ?    Which,  metal  is  the  most 
malleable  ? 

164.  Explain  brittleness.     In  what  sense  is  a  sheet  of 
glass  stronger,  and  in  what  sense  is  it  weaker,  than  a  sheet 
of  paper  ? 

165.  Define  hardness.     If  we  have  three  bodies  A,  B, 
and  C,  how  are  their  proper  places  on  a  scale  of  relative 
hardness  found  ? 

166.  Which  is  the  hardest  of  all  known   substances, 
and  how  is  it  cut  ? 

167.  Explain  the  processes   known  as  tempering  and 
annealing.     What  are  Prince  Rupert's  drops  ? 

168.  What  does  the  word  elasticity  denote  ?    What  is 
meant  by  the  limit  of  perfect  elasticity  ? 

169.  What  condition  should  a  solid  structure  such  as 
a  bridge  fulfil  ? 

170.  Give  the  laws  which  hold  true,  within  the  limit 
of  perfect  recovery,  of  resistance  to  linear  extension. 

171.  What  relation  exists  between  the  forces  which 
will  produce  equal  amounts  of  linear  extension  and  linear 
compression  respectively  in  the  same  body  ? 

172.  How  are  the  forces  with  which   different   sub- 
stances resist  linear  extension  or  compression  compared 
together  ? 

173.  How   are   experiments   on    torsion    conducted  ? 
What  laws  have  been  established  ? 

174.  Mention  some  of  the  ways  in  which  the  force  with 
which  a  solid  resists  any  attempt  to  bend  it  is  utilized. 

175.  What  are  the   laws   of  flexure  ?    What  follows 
from  the  last  law  as  to  the  best  form  for  a  beam  of  given 
mass  which  is  to  be  heavily  loaded  ? 


LESSON  XI.  —  Forces  exhibited  in  Liquids. 

176.  What  is  the  essential  difference  between  a  solid 
and  a  liquid  ?  What  are  proofs  that  cohesion  is  not  entire- 
ly wanting  in  liquids  ? 


LESS,  xi.]          ELEMENTARY  PHYSICS.  17 

177.  Describe   the   state    of  liquidity    called    viscous. 
Give    instances   to   show    that    time   is   an   element   of 
importance  in  determining  the  liquidity  of  a  substance. 

178.  What  is  characteristic  of  the  resistance  to  compres- 
sion offered  by  liquids  ?     What  is  the  exact  measure  of 
the  compressibility  of  water  ? 

179.  State  the   law  of  liquid  pressure  discovered  by 
Pascal,  and  illustrate  it  by  the  imaginary  case  of  a  hollow 
vessel  full  of  water  which  is  uninfluenced  by  gravity. 

180.  Show  that,    by   employing   pistons   of  different 
sizes,  a  fluid  is  capable  of  forming  a  very  powerful  mechan- 
ical arrangement. 

181.  What  is  the  principle,  and  what  are  some  of  the 
uses,  of  Bramah's  press  ] 

182.  Prove  that  the  surface  of  a  liquid  in  an  open  ves- 
sel must  be  perpendicular  to  the  force  of  gravity,  that  is, 
horizontal.     What  is  the  true  character  of  the  surface  of  a 
large  body  of  water,  as  the  ocean  1 

183.  Explain  the  construction  and  use  of  the  water- 
level. 

184.  Explain   the  construction  and  use  of  the  spirit- 
level. 

185.  Explain  Artesian  wells. 

186.  What  is  the  measure    of  the  pressure   on   the 
horizontal  base  of  an  open  vessel  full  of  a  liquid  ?     Hence 
show  what  must  be  the  relation  between  the  pressure  on 
any   horizontal  layer   of  liquid  in  an  open  vessel  and 
(1)  the  depth  of  the  layer,  (2)  the  area  of  the  layer. 

187.  Show  by  a  simple  experiment  that  the  pressure 
of  a  horizontal  layer  of  water  is  the   same    upwards  as 
downwards. 

138.  A  hollow  cubic  decimetre,  open  at  the  top,  is 
filled  with  water ;  what  will  be  the  pressure  on  the 
bottom  and  sides  ] 

189.  A  vessel  contains  water  to  the  depth  of  a  deci- 
metre, and  one  of  the  sides  of  this  vessel  is  a  rectangular 
surface,  the  bottom  of  which  is  one  decimetre,  while  the 


18  QUESTIONS  ON  STEWART'S      [CHAP.  n. 

side  slopes  at  an  angle  of  45°  ;  what  is  the  whole  press- 
ure on  this  side  ? 

190.  In  what  way  is  the  pressure  exerted  by  a  liquid 
connected  with  the  density  of  the  liquid  1 

191.  Prove  that  a  fluid  buoys  up  a  solid  immersed  in 
it  with  a  force  equal  to  the  weight  of  the  fluid  displaced. 

192.  Apply  the  principle  of  buoyancy  successively  to 
the  cases  in  which  the   density  of  the   solid  immersed 
is  greater  than,  equal  to,  and  less  than  that  of  the  fluid. 

193.  A  cube  of  wood,  the  density  of  which  is  equal  to 
0*8,  is  put  into  a  vessel  containing  water ;  what  portion 
of  its  side  will  be  immersed  ? 

194.  Taking  water  at  4°  C.  as  the  standard  of  specific 
gravities  or  relative  densities,  how  do  you  define  the  spe- 
cific gravity  of  any  substance  ?     If,  as  in  the  Metric  sys- 
tem, the  density  of  water  at  4°  C.  is  equal  to  unity,  what 
relation  must  exist  between  the  specific  gravity  and  the 
density  of  any  substance  1 

195.  Explain  a  method  of  finding  the  specific  gravity 
of  a  solid  body.     Suppose,  for  the  sake  of  illustration, 
that  a  substance  weighs  in  vacuo  120  grammes,  and  when 
immersed  under  water  at  4°  C.  only  89  grammes ;  find  the 
specific  gravity  of  the  substance.    Hence,  what  general  rule 
may  be  given  for  obtaining  the  specific  gravity  of  a  solid 
substance  1 

196.  Explain  a  method   of  ascertaining  the   specific 
gravity  of  a  liquid.     As  an  illustrative  example,  let  the 
loss  of  weight  of  a  solid  body  in  water  equal  31  grammes, 
and  its  loss  of  weight  in  the  liquid  in  question  equal  28 
grammes.     Find  the  specific  gravity  of  the  liquid. 

197.  Describe  some  capillary  phenomena  which  show, 
(1)  what  two  kinds  of  capillary  action  exist,  (2)  what  in- 
fluence the  diameter  of  the  tube  has  on  capillary  ascent 
and  depression. 

198.  Mention  other  illustrations  of  the  laws  of  capil- 
larity. 

199.  Describe  endosmose  and  eooosmose. 


LESS,  xii.]        ELEMENTARY  PHYSIOS.  19 

LESSON  XII.  —  Forces  exhibited  in  Gases. 

200.  In  what  respect  does  a  gas  differ  from  a  liquid  ? 
In  what  respects  is  it  like  all  other  substances  ? 

201.  Describe  an  experiment  illustrating  that  a  gas  has 
weight,  and  also  that  some  gases  weigh  more  than  others. 

202.  How  may  a  liquid  be  converted  into  a  gas  1 

203.  True  steam  being  invisible,  how  do  you  account 
for  the  visible  cloud  arising  from  a  kettle  or  a  railway 
engine  1 

204.  State  and  illustrate  the  distinction  between  gases 
and  vapors. 

205.  What  agencies  tend  to  bring  a  gas  into  the  liquid 
or  solid  state  1    What  six  gases  have  never  yet  been  .li- 
quefied ?    What  substance  has  never  yet  been  vaporized  ? 

206.  What  is  the  composition  of  the  atmosphere  1 

207.  What  effect  on  the  air  would  the  processes  of 
respiration  and  combustion,  if  unbalanced,  in  course  of 
time  produce  I     Why  does  the  composition  of  the  atmos- 
phere in  point  of  fact  remain  unchanged  1 

208.  Why  is  it  that,  although  the  air  is  exerting  press- 
ure all  around  us,  we  seldom  perceive  any  traces  of  it  ? 

209.  What  is  the  experiment  of  the  Magdeburg  hemi- 
spheres, and  what  does  it  illustrate  1 

210.  How  was  the  ascent  of  water  in  pumps  accounted 
for  up  to  the  time  of  Galileo  1     Give  Torricelli's  reason- 
ing, together  with  his  celebrated  and  decisive  experiment, 
upon  this  subject. 

211.  How  was  the  truth  of  Torricelli's  discovery  veri- 
fied by  Pascal  ? 

212.  What  remarks  are  made  by  the  author  in  regard 
to  the  connection  between  the  barometer  and  the  weather  ? 

213.  If  we  cork  a  flask  full  of  air  and  then  remove 
half  of  the  mass  of  air  within  the  flask,  how  much  will 
the  pressure  on  the   interior  of  the  flask   be  changed  ? 
What  is  that  statement  of  Boyle's  law,  which  gives  di- 
rectly the  relation  between  the  mass  and  the  pressure  of 
a  quantity  of  gas  1 


'^20  QUESTIONS  ON  STEWART'S      [CHAP.  n. 

214.  State  the  law  of  Boyle  in  the  form  adopted  hy 
the  discoverer,  and  verify  its  truth  by  a  simple  experi- 
ment. 

215.  Show  that  the  two  forms  of  stating  Boyle's  law 
come  to  the  same  thing. 

216.  What  is  the  true  explanation  of  gaseous  pressure 
according  to  many  philosophers  ?     Show,  with  the  aid  of 
a  numerical  illustration,  that  their  hypothesis  is  in  har- 
mony with  Boyle's  law. 

217.  Show,  by  an  experiment,  that  gases   as  well    as 
liquids  possess  buoyancy,  and  are  subject  to  the  principle 
of  Archimedes. 

218.  What  is  the  reason  that  a  "balloon  rises  in  the  at- 
mosphere ? 

219.  Describe  the  construction  and  action  of  an  air- 
pump. 

220.  Prove  that  the  density  of  the  air  in  the  receiver 
diminishes   in   a  geometrical  ratio,  i.  e.  that   a   constant 
fractional  part  of  the  mass  of  air  remaining  in  the  receiver, 
is  expelled  by  each  double  stroke.    Why  could  we  never, 
even  in  theory,  succeed  in  producing  a  perfect  vacuum '? 
When  is  a  practical  limit  reached  ? 

221.  Describe  the  construction  and  action  of  the  com- 
mon lifting-pump.      What   sets  a  limit  to  the  height  to 
which  water  can  be  raised  by  means  of  this  pump  ? 

222.  Describe  a  siphon,  and  its  action,  and  explain  why 
the  flow  of  liquid  from  one  vessel  to  the  other  is  main- 
tained.    When  will  a  siphon  once  set  in  action  cease  to  act  ? 
What  fixes  a  limit  to  the  working  length  of  the  shorter 
arm  of  a  siphon  ? 

223.  Describe  Graham's  experiment  on  gaseous  diffu- 
sion. 

224.  Mention   substances   which   have   the   power   of 
absorbing  or  retaining  matter  in  the  gaseous  state. 


LESS,  xiii.]       ELEMENTARY  PHYSICS.  21 


CHAPTER  III. 

ENERGY. 

LESSON  XIII.  —  Definition  of  Energy. 

225.  Adduce  examples  which  illustrate   the   applica- 
tion, up  to  recent  times,  of  Newton's  law  of  action  and 
reaction.      What  view  did  the  old  hypothesis  take  of  the 
phenomena  of  collision  and  friction  ? 

226.  Mention  the  considerations  which  probably  led 
the  way  to  a  numerical  estimate  of  work. 

227.  Define  the  unit  of  work,  and  show  how  to  find 
the  amount  of  work  done  in  lifting  a  body  to  any  given 
height. 

228.  Define   energy,  and  estimate  how  much  energy 
will  be  imparted  to  a  stone  weighing  one  kilogramme  by 
projecting  it  vertically  upwards  with  a  velocity  of  9 '8 
metres  per  second. 

229.  Prove  that  the  work  which  can  be  accomplished 
by  a  moving  body  is  proportional,  (1)  to  the  square  of  its 
velocity,  (2)  to  its  mass. 

230.  Show  that  the  work  capable  of  being  done  by  a 
body   whose   mass   (in   kilogrammes)   is   m,   and   whose 
velocity  (in  metres)  is  v,  is  represented  by  the  expression 

m  t<2  1  ..    ~  .     m  i&  . 

j^-.     (The  general  form  is  -a~.) 

231.  Estimate    the   energy   of    a  body   weighing    64 
grammes  projected  vertically  upwards  with   the  velocity 
of  60  metres  per  second. 

232.  Show,  by  means  of  the  force  of  gravitation,  that 
two  types  or  kinds  of  energy  exist,  which  are  mutually 
convertible. 


22  QUESTIONS  ON  STEWART'S    [CHAP.  in. 

LESSON  XIV.  —  Varieties  of  Energy, 

233.  Give  a  summary  of  what  was  said  about  the  two 
forms  of  energy  in  the  last  Lesson. 

234.  Illustrate  the   transformation   of  energy  by  an 
example,  taken  from  chemistry,  and  point  out  the  analo- 
gies to  the  case  of  gravitation.     What  appears  from  this 
example  to  be  the  true  nature  of  heat  1 

235.  Name  some  of  the  varieties  of  visible  or  mechani- 
cal energy,  both  kinetic  and  potential. 

236.  How  does  the  doctrine  that  heat  is  a  form  of 
energy  explain  the  phenomena  of  latent  heat  1 

237.  Trace  the  analogy  between  the  mechanical  world 
and  tlje  molecular  world  by  means   of  the  phenomena 
of  sound  and  of  radiant  light  and  heat. 

238.  Show  that  electrical  phenomena  afford  another 
illustration  of  molecular  energy. 

239.  What  great  advantage  as  a  source  of  energy  does 
electricity  in  motion  possess  over  water  in  motion  or  heat  ? 

240.  Give  a  brief  recapitulation  of  the  various  forms 
of  energy. 

LESSON  XV.  —  Conservation  of  Energy. 

241.  What  was  formerly  in  many  minds   the  great 
ideal  of  mechanical  triumphs  ? 

242.  What  conclusive  answer  may  now  be  made  to  the 
arguments  in  favor  of  perpetual  motion  ? 

243.  What  is  the  principle   of   the   conservation   of 
energy,  and  what  is  the  nature  of  the  evidence  in  favor  of 
its  truth  ? 

244.  Apply  the  principle  of  the  conservation  of  energy 
to  the  case  of  a  stone  projected  vertically  upwards. 

245.  What  becomes  of  the  energy  of  a  railway  train 
when  it  is  suddenly  stopped  ?  or  of  a  cannon-ball  after  it 
has  struck  the  target  ?  or,  in  general,  what  becomes  of  the 
energy  of  visible  motion  when  it  has  been  stopped  by  per- 
cussion or  friction  ? 


LESS,  xv.]         ELEMENTARY  PHYSICS.  23 

246.  What  experimental  evidence  upon  this  point  do 
we  owe  to  Rumford  and  Davy  ?     What  simple  phenomena 
can  you  mention  which  furnish  evidence  in  the  same  di- 
rection ? 

247.  Who  were  the  first  to  point  out  the  probability 
of  a  connection  between  the  various  forms   of  energy? 
Who  established  this  connection  on  a  scientific  basis  ? 
In  particular,  what  was  the  result  of  the  researches  of 
Joule  ? 

248.  Illustrate  still  further  the  connection  between  the 
various  kinds  of  energy  by  what  takes  place  in  a  galvanic 
battery. 

249.  Give  an  algebraic  statement  of  the  doctrine  of  the 
conservation  of  energy. 

250.  Illustrate  the  true  function  of  a  machine,  by  ap- 
plying the  law  of  the  conservation  of  energy  to  one  of  the 
ordinary  mechanical  combinations,  such  as  a  system  of 
pulleys. 

251.  Apply  the  same  law  to  the  hydraulic  press. 

252.  What  law  holds  universally  in  machines,  if  fric- 
tion be  left  out  of  account,  respecting   the   power,  the 
weight,  and  the  distances  which  they  traverse  2 


24  QUESTIONS  ON  STEWART'S      [CHAP.  iv. 


CHAPTER  IV. 

VISIBLE   ENERGY  AND   ITS   TRANSMUTATIONS. 

LESSON  XVI.  —  Varieties  of  Visible  Energy. 

253.  Mention  some  of  the  varieties  of  visible  kinetic 
energy. 

254.  Why  is  the  energy  of  a  vibrating  string  classed 
among  the  forms  of  visible  energy  ? 

255.  Give  examples  of  visible  potential  energy. 

256.  Suppose   a   rifle-ball  weighing  20  grammes  and 
moving  with  a  velocity  of  200  metres  per  second  buries 
itself  in  a  block  of  wood  weighing  20  kilogrammes  and 
hung  by  a  string  so  as  to  form  a  pendulum.     Find  how 
much  of  the  energy  of  the  ball  will  reappear  after  the  im- 
pact in  the  form  of  visible  energy.     What  has  become  of 
the  remainder  of  the  energy  of  the  ball  1 

257.  What  distribution  of  momentum,  and  what  con- 
sequent change  of  energy,  take  place  during  the  passage 
of  a  ball  through  the  air  ? 

258.  What  extension  of  the  first  law  of  motion  are  we 
now  prepared  to  recognize  1 

259.  Examine  the  following  case  of  direct  impact  of 
two  inelastic  solids,  and  find  what  part  of    the   united 
energy  of  the  two  masses  before  impact  is  transmuted  into 
heat  :  —  weights    of    the    solids,    20    grammes    and    10 
grammes;  velocity  of  the  first,  20  ;  of  the  second,  16  in 
an  opposite  direction. 

260.  What  is  the  law  of  energy  which  applies  to  the 
case  of  the  impact  of  two  perfectly  elastic  bodies  ?     Sup- 
pose, as  an  illustrative  example ,  that  two  perfectly  elastic 
balls,  weighing  respectively  4  and  3  kilogrammes,  moving 
in  the  same  direction  with  velocities  5  and   4,  impinge 
on  each  other. 


LESS,  xvi.]        ELEMENTARY  PHYSICS.  25 

261.  Apply  the  laws  of  impact  to  the  case  in  which 
one  elastic  ball  impinges  directly  against  the  extremity 
of  a  row  of  elastic  balls  at  rest. 

262.  Consider  briefly  the  energy  of  a  circular  disk  in 
rapid  rotation.     In  general,  wherever  in  nature  a  body 
revolves  in  a  circle  round  a  central  force,  what  is  true  of 
its  velocity  and  of  its  kinetic  energy  ? 

263.  Consider  the   energy  of  a   body  moving  in   an 
ellipse,  and  show  that  a  definite  part  of  its  energy  assumes 
alternately  the  kinetic  and  the  potential  forms. 

264.  Show  that  the  doctrine  of  energy  leads  to  the 
conclusion  that  the  velocity  of  a  body  which  has  slid 
down  a  smooth  inclined  plane  depends  only  on  the  height 
of  the  plane.     Why  does  this  proposition  fail  in  case  the 
plane  be  rough  ? 

265.  Mention  some  instances  of  visible  energy  of  posi- 
tion, and  state  how  the  energy  in  each  case  may  be  con- 
verted into  energy  of  motion. 

266.  Show  that  the  energy  of  an  oscillating  pendulum 
is  alternately  kinetic  and  potential. 

267.  A  pendulum  bob  weighing  one  kilogramme  is  so 
swung  that  it  is  higher  at  the  summit  of  its  oscillation 
than  at  its  lowest  point  by  one  decimetre  ;   what  is  its 
velocity  at  its  lowest  point  1 

268.  Describe  Foucault's  pendulum  experiment,  the 
pendulum  being  supposed,  for  the  sake  of  simplicity,  to 
oscillate  at  the  north  pole, 

269.  Consider  the  energy  of  a  vibrating  body,  as  the 
string  of  a  musical  instrument  or  a  bell,  and  point  out  the 
analogies  between  the  motion  of  such  a  body  and  that 
of  a  pendulum. 

270.  What  effect   does  vibratory  motion   in  the  air 
usually  produce  before  it  assumes  the  shape  of  heat  ? 

271.  Give   a  recapitulation  of  the  various   kinds   of 
visible  energy  which  have  been  considered  in  this  Les- 


26  QUESTIONS  ON  STEWART'S     [CHAP.  iv. 

LESSON  XVII.  —  Undulations. 

272.  Prove  that  for  small  vibrations  of  a  pendulum 
the  force  which  urges  the  ball  in  the  direction  of  its  mo- 
tion at  each  instant  is  proportional  to  the  distance  of  the 
ball  from  its  point  of  rest,  i.  e.  is  proportional  to  the  dis- 
placement. 

273.  Explain  the  principle  of  isochronism.     How  far 
is  it  applicable  ? 

274.  Upon  what  elements  does  the  time  of  vibration 
of  a  body  depend  ?     How  may  this  be  illustrated  ? 

275.  Define  wave  motion,  and  give  examples  of  it. 

276.  Define  an  up-and-down  or  transverse  wave,  and  a 
wave-length;   and  illustrate   by  diagrams  the  manner  in 
which  an  undulatory  motion  is  propagated  by  an  up-and- 
down  wave. 

277.  If  v  denote  the  velocity  with  which  the  wave  is 
propagated,  t  the  time  of  a  double  vibration  of  a  particle, 
and  I  a  wave-length,  show  that  I  =  v  t. 

278.  What  is  the  nature  of  waves  of  condensation  and 
rarefaction,  or  longitudinal  waves  ? 

279.  What  is  meant  by  the  phase  of  a  vibrating  par- 
ticle ? 

280.  What  is  an  essential  peculiarity  of  vibrating  mo- 
tion as  regards  the  phases  of  any  two  contiguous  parti- 
cles ? 

281.  What  is  the  amplitude  of  a  vibration  1 

282.  Explain  why  the  wave-length  does  not  depend 
on  the  amplitude  of  the  vibration. 


LESSON  XVIII.  —  Sound. 

283.  What  is  the  definition  of  Acoustics  ? 

284.  In  what  two  senses  is  the  word  "sound"  used] 
In  which  sense  is  it  here  used  1 

285.  Explain  the  difference  between  a  noise  arid  a  mu- 
sical sound  or  note. 

286.  What  determines  the  pitch  of  a  note  1 


LESS,  xviii.]    ELEMENTARY  PHYSICS.  27 

287.  What  other  characteristics  of  sound  besides  pitch 
are  perceived  by  the  ear  ? 

288.  What  is  the  nature  of  sound-waves  1 

289.  How  may  it  be  shown  that  sound  is  not  prop- 
agated in  vacuo  1 

290.  What  are  the  laws  of  the  reflection  of  sound? 
Illustrate  them  by  means  of  a  diagram. 

291.  What  constitutes  an  echo  ?    When,  for  instance, 
is  an  echo  more  audible  than  the  original  sound  ? 

292.  What  condition  must  be  fulfilled  in  order  to  pro- 
duce a  distinct  echo  1    What  effect  do  whispering-galleries 
have  upon  a  sound  ? 

293.  What  experiment  upon  sound  may  be  performed 
with  two  conjugate  reflectors  1 

294.  What  is  related  of  the  Cathedral  of  Girgenti  in 
Sicily  ? 

295.  What  power  does   a   convex  glass  lens  exercise 
upon  rays  of  light  ? 

296.  What  was  the  experiment  of  M.  Sondhauss,  and 
the  conclusion  to  be  drawn  from  it  1 

297.  What  is  the  velocity  of  sound  in  air,  and  by  what 
method  was  it  determined  ? 

298.  What   effect,   if  any,   have  pitch  and  intensity 
respectively  on  the  velocity  of  sound  ? 

299.  How  does  the  velocity  of  sound  in  air  compare 
with  its  velocity  in  hydrogen  gas  at  the  same  pressure  ? 
What  reason  can  be  given  for  this  difference  2 

300.  Explain  why  the  velocity  of  sound  in  a  given 
medium  does  not  vary  wTith  its  density. 

301.  Why  does  sound  travel  faster  in  warm  than  in 
cold  air  1 

302.  What  is  the  velocity  of  sound  in  water  ?  in  wood  1 

303.  Prove  that  the  intensity  of  a  sound  varies  inverse- 
ly as  the  square  of  the  distance  from  the  source. 

304.  Why  is  it  probable  that  the  intensity  of  sound 
really  diminishes  somewhat  more  rapidly  than  is  indi- 
cated by  the  law  of  "  inverse  squares  "  ? 

305.  What  effect  does  a  change  in  the  density  of  the 


28  QUESTIONS  ON  STEWART'S     [CHAP.  iv. 

medium  have  upon  the  intensity  of  sound  ?     How  may 
this  law  be  verified  1    What  is  the  reason  that  it  is  true  'I 

306.  Why  is  sound  much  better  heard  when  the  air  is 
calm  and  homogeneous  ? 

307.  By  what  arrangement  may  the  intensity  of  the 
sound  of  a  musical  string  be  strengthened  ? 


LESSON  XIX.  —  Vibrations  of  Sounding  Bodies. 

308.  State  the  laws  which  connect  the  vibrations  of  a 
stretched  string  -with  the  properties  of  the  string. 

309.  Explain  the  mode  of  action  of  an  organ  pipe. 

310.  What  is  the  reason  that  an  organ  pipe  filled  with 
any  other  gas  than  air  yields  an  entirely  different  sound  1 
What  use  may  be  made  of  this  fact  ? 

311.  What  acoustical   difference  is  there  between  a 
shut  pipe  and  an  open  pipe  of  the  same  dimensions  1 

312.  Suppose  we  have  rods  of  wood  fixed  at  one  end 
and  free  to  move  at  the  other  ;  wrhat  two  kinds  of  vibra- 
tory motion  may  be  imparted  to  them  ? 

313.  What  are  the  laws  which  express  the  relations 
between  the  number  of  vibrations  of  a  rod  and  its  dimen- 
sions ? 

314.  What  is  the  law  which  governs  the  vibrations  of 
plates  1 

315.  What  are  nodal  points  or  nodes  on  a  vibratory 
cord,  and  how  may  they  be  produced  ?    What  is  a  loop  or 
ventral  segment  ? 

316.  In  producing  nodes  along  a  cord,  what  condition 
must  be  observed  in  order  that  the  vibrations  shall  not 
interfere  Avith  each  other  ? 

317.  How  may  the  existence  of  nodes  and  ventral  seg- 
ments be  rendered  evident  by  means  of  vibratory  plates  1 

318.  What  law  governs  the  communication  of  vibra- 
tions from  one  instrument  to  another  through  the   air  1 
Mention  instances  in  which  this  communication  may  be 
observed. 


LESS,  xix.j       ELEMENTARY  PHYSICS.  29 

319.  Explain  the  action  of  Savart's  machine  for  meas- 
uring the  number  of  vibrations  corresponding  to  a  given 
sound. 

320.  Describe  Lissajous's  method  of  making  vibrations 
apparent  to  the  eye. 


30  QUESTIONS  ON  STEWART'S      [CHAP.  v. 


CHAPTER   V. 

HEAT. 

LESSON  XX.  —  Temperature. 

321.  What  two  kinds  of  energy  are  denoted  by  the 
word  heat  ? 

322.  What  three-fold  division  of  the  study   of  phe- 
nomena  connected  with   heat  will  be   adopted  in  this 
chapter  ? 

323.  When  are  two  bodies  said  to  be  of  the  same  tem- 
perature ?    When  is   one  body  said  to  be   of    a  higher 
temperature  than  another  1     How  would  you  define  tem- 
perature ? 

324.  What  is  the  general  law  of  expansion  under  the 
action  of  heat  1     Cite  experiments  which  go  to  prove  this 
law. 

325.  What  is  one  remarkable  exception  to  the  above 
law  of  expansion  ? 

326.  Why  in  general  does  expansion  fail  to  furnish  us 
with  an  exact  measure  of  temperature  ?     How  does  the 
case  of  water  illustrate  this  ? 

327.  Why  is  a  substance  near  the  point  of  changing  its 
state  not  fitted  to  be  used  as  a  means  of  measuring  tem- 
perature ? 

328.  Compare  the  relative  merits  of  mercurial  and  air 
thermometers  as  regards  accuracy  and  convenience. 

329.  What  is  the  principle  of  the  mercurial  thermome- 
ter?    Describe  the  process  of  filling  the  bulb  and  tube 
with  mercury. 

330.  How  may  great  delicacy  be  secured  in  a  ther- 
mometer ? 

331.  What  are  the  two  fixed  points  of  temperature  on  a 
thermometer  ?    Describe  how  they  are  determined. 


LESS,  xx.]         ELEMENTARY  PHYSICS.  31 

332.  Explain  how  the  stem  of  a  thermometer  is  gradu- 
ated according  to  the  centigrade  scale.     How  do  the  Fah- 
renheit and  the  Reaumur  scales  differ  from  the  Centi- 
grade 1 

333.  Deduce  formulae  for  reducing  Centigrade  degrees 
to  Fahrenheit,  and  vice  versa. 

334.  Find  the  degree  of  Fahrenheit  which  corresponds 
to  45°  Centigrade. 

335.  Find  the  degree  Centigrade  which  corresponds  to 
86°  Fahrenheit. 

336.  What  degree  Fahrenheit  corresponds  to  40°  Centi- 
grade ? 

337.  Deduce  formulae  for  reducing  Centigrade  degrees 
to  Keaumur,  and  vice  versa. 

338.  Explain  the  change  of  zero  of  a  mercurial  ther- 
mometer, and  the  corrections  which  must  be  applied  to 
eliminate  this  source  of  error. 

339.  What  is  the  effect  of  suddenly  heating  and  cooling 
a  thermometer  ?     What  practical  rule  does  this  effect  sug- 
gest with  regard  to  the  determination  of  the  fixed  points  ? 

340.  Explain  why  it  may  be  necessary  to  apply  a  cor- 
rection to  the  reading  of  a  mercurial   thermometer  on 
account  of  the  position  in  which  the  instrument  is  held. 

341.  Suppose  a  mercurial  thermometer  is  placed  under 
the  receiver  of  an  air-pump  and  the  air  exhausted,  what 
will  be  the  effect  on  the  column  of  mercury  1 

342.  If  the  bulb  of  a  thermometer,  together  with  the 
lower  part  of  the  stem  up  to  the  zero-point,  be  immersed 
in  boiling  water,  while  the  remainder  of  the  stem  from  0° 
to  100°  is  exposed  to  air  at  the  freezing-point,  the  temper- 
ature denoted  will  be  98'4°  instead  of    100°.     Explain 
this. 

343.  Why  is  mercury  unsuited  for  measuring  very  low 
temperatures  1     What  liquid  is  used  for  this  purpose  1 

344.  Describe  a  minimum  thermometer  and  its  action. 

345.  Describe  Phillip's  maximum  thermometer  and  its 
action. 

346.  Describe  Leslie's  differential  thermometer  and  its 
action. 


32  QUESTIONS   ON  STEWART'S      [CHAP.  v. 

LESSON  XXI.  —  Expansion  of  Solids  and  Liquids  through 
Heat. 

347.  Explain  (with,  a  diagram)  how  the  comparatively 
small  expansion  of  a  solid  rod  due  to  heat  may  be  rendered 
visible  by  mechanical  means.     What  other  method   of 
magnifying  expansion  is  also  employed  ? 

348.  Define  the  coefficient  of  linear  expansion  of  a  sub- 
stance. 

349.  Which  is  the  most  expansible  of  the  metals  men- 
tioned in  the  table  ?  which  the  least  1 

350.  What  circumstances  may  modify  the  value  of  the 
coefficient  of  expansion  of  a  substance  ?     Give  an  exam- 
ple. 

351.  Show  that  the  cubical  expansion  of  a  substance 
is  very  nearly  three  times  as  great  as  the  linear  for  the 
same  change  of  temperature. 

352.  State  the  principle  of  one  method  of  determining 
experimentally  the  cubical  expansion  of  solids. 

353.  Illustrate  this  method  by  solving  the  following 
example  :  — 

A  solid  weighs  600  grammes  in  vacuo,  and  only  400 
grammes  in  a  fluid  at  0°  C.,  of  which  the  specific  gravity 
is  1*2,  while  it  weighs  406  grammes  in  the  same  fluid  at 
100°  C.,  for  which  temperature  the  specific  gravity  of  the 
fluid  is  known  to  be  1*16  ;  find  the  cubical  expansion  of 
the  solid  between  these  two  temperatures. 

354.  What  general  relation  between  the  coefficients  of 
linear  and  cubical  expansion  of  any  substance  is  indicated 
by  the  table  on  page  161  ? 

355.  What  notable  exceptions  are  there  to  the  follow- 
ing laws  :  — 

(1)  Solids  expand  through  heat, 

(2)  And  expand  equally  in  all  directions. 

356.  What  effect,  in  general,  has  the  temperature  of  a 
solid  on  its  rate  of  expansion  ? 

357.  Explain  the  distinction  between  the  apparent  and 
the  real  expansion  of  a  liquid. 


LESS,  xxii.]      ELEMENTARY  PHYSICS.  33 

358.  Give   an  outline  of  the  method  of  finding  the 
real  expansion  of  a  liquid,  called  the  method  by  ther- 
mometers. 

359.  Explain  Matthiesson's  method  of  determining  the 
real  expansion  of  a  liquid. 

360.  Illustrate  Matthiesson's   method  by  solving  the 
following  example  :  — 

A  piece  of  glass,  of  which  the  linear  expansion  from  0° 
to  100°  C.  is  known  to  be  0.0009,  loses  at  0°  C.  one  gramme 
of  its  weight  in  the  fluid  in  which  it  is  weighed,  while  at 
100°  C.  it  only  loses  0.96  of  a  gramme.  Find  the  expan- 
sion of  the  fluid  between  0°  and  100°  C. 

361.  Give  an  outline  of  Regnault's  method  of  deter- 
mining the  expansion  of  mercury. 

362.  What  peculiarity  does  water  exhibit  with  respect 
to  its  expansion? 

363.  Explain  Hope's  method  of  ascertaining  the  point 
of  maximum  density  of  water. 

364.  What  fact  respecting  the  rate  of  expansion  of 
water  at  different  temperatures  is  shown  by  the  table  on 
page  165  ? 

365.  What  reasons  do  we  have  for  believing  that  the 
rate  of  expansion  of  very  volatile  liquids  must  be  very 
great  ?     How  is  this  conclusion  verified  in  the  case   of 
liquid  carbonic  acid  ? 

366.  Recapitulate  the  chief  laws  for  the  expansion  of 
liquids. 


LESSON  XXII.  —  Expansion  of  Gases,     Practical  Appli- 
cations, 

367.  State  and  illustrate  the  law  discovered  by  Charles, 
which  expresses  the  relation  between  the  temperature  and 
the  pressure  of  a  gas,  the  volume  remaining  constant. 

368.  Prove  that  the  same  law  expresses  also  the  rela- 
tion between  the  temperature  and  the  volume  of  a  gas, 
the  pressure  remaining  constant. 

2*  o 


34  QUESTIONS   ON  STEWART'S      [CHAP.  v. 

369.  A  bladder  which  at  0°  contains  900  cubic  cen- 
timetres of  air  has  its  temperature  increased  to  30°  C.,  the 
pressure  under  which  the  gas  exists  meanwhile  remaining 
constant ;  what  will  now  be  the  volume  of  the  gas  in  the 
bladder  ? 

370.  What  are  some  direct  consequences  of  the  fact  that 
the  coefficient  of  expansion  is  the  same  for  all  gases  ? 

371.  Explain  more  fully  than  was  possible  in  Lesson 
XX.  the  advantage  of  using  an  air  thermometer. 

372.  At  what  temperatures  are  the  metre  and  the  yard 
respectively  standards  of  length  ? 

373.  Explain  why  it  is  necessary  in  all  very  accurate 
weighings  to  know  the  temperature  of  the  air. 

374.  Show  that,  in  the  metric  system,  the  weight  (in 
grammes)  of  one  cubic  centimetre  of  any  substance  will 
denote  at  the  same  time  its  specific  gravity. 

375.  Why  is  it  necessary  to  fix  upon  a  standard  temper- 
ature in  comparing  the  specific  gravities  of  substances, 
and  what  is  this  standard  1 

376.  What  is  the  standard  pressure  employed  in  com- 
paring the  specific  gravities  of  gases  ? 

377.  Explain  what  effect  change  of  temperature  will 
produce  upon  the  motion  of  a  pendulum,  and  also  upon 
the  motion  of  the  balance-wheel  of  a  chronometer. 

378.  Explain  Harrison's  gridiron  pendulum. 

379.  Explain  the  compensation  balance  for  chronome- 
ters. 

380.  Mention  other  instances  in  which  account  must 
be  taken  of  the  expansion  of  bodies. 

381.  Mention  instances  in  which  advantage  is  taken  of 
the  fact  of  expansion. 


LESSON  XXIII.  —  Change  of  State  and  other  Effects  of  Heat 

382.  What  invariable  rule  holds  in  the  production  of 
changes  of  state  through  heat  ? 

383.  Give  examples  of  substances  which  differ  from 


LESS,  xxiii.]     ELEMENTARY  PHYSICS.  35 

one  another  in  the  manner  of  their  passage  from  the  solid 
to  the  liquid  state.     What  is  this  passage  called  1 

384.  Give  instances  in  which  change  of  composition 
accompanies  change  of  state. 

385.  What  effect  does  pressure  have  on  the  melting- 
point  of  ice  1    What  general  law  is  found  to  hold  true 
with  respect  to  the  connection  between  pressure  and  con- 
gelation 1 

386.  What  is  the  reason  that  gold,  silver,  and  copper 
coins  cannot  be  cast  in  a  mould,  but  must  be  stamped  ? 

387.  Under  what  conditions  may  water  be  cooled  be- 
low the  freezing-point  without  becoming  ice  1     What  other 
instances  of  a  similar  phenomenon  exist  ?     In  all  such 
cases  how  may  solidification  be  immediately  produced  ? 

388.  What  is  regelation,  arid  what  is  Forbes's  explana- 
tion of  it  ? 

389.  What  is  the  difference  between  sublimation  and 
vaporization  1 

390.  Distinguish  between  two  kinds  of  vaporization. 

391.  What  did  Daltoii  show  with  respect  to  the  forma- 
tion of  a  vapor  in  a  confined  space  1 

392.  How  far  is  the  evaporation  of  a  liquid  modified 
by  taking  place  in  air  instead  of  in  vacuo  ? 

393.  What  conditions  are  most  favorable  to  rapid  evap- 
oration in  the  open  air,  and  why  ? 

394.  Explain  the  apparatus   and  process  of  distilla- 
tion. 

395.  What  is  ebullition,  and  how  is  it  affected  by  the 
supply  of  heat  ? 

396.  Enumerate  the  chief  circumstances  upon  which 
the  boiling-point  of  a  liquid  depends. 

397.  What  are  the  boiling-points  of  ether  and  of  mer- 
cury, as  given  in  the  table  on  page  1 78  ? 

398.  Describe   two  simple   experiments  which    show 
that  the  boiling-point  of  a  liquid  depends  upon  the  press- 
ure under  which  it  exists. 

399.  Why  is  the  boiling-point  of  water  at  the  top  of  a 
mountain  lower  than  at  its  bottom  1    How  does  this  fact 


36  QUESTIONS  ON  STEWART'S      [CHAP.  v. 

interfere  with  culinary  operations,  and  how  is  the  diffi- 
culty remedied  ? 

400.  In  what  way  is  the  fact  that  the  boiling-point  of 
water  is  lessened  as  we  rise  above  the  sea-level  applied  tor 
a  practical  use  1 

401.  How  do  glass  and  metal  vessels  compare  with 
each  other  in  their  influence  upon  the  boiling-point  of 
water  ?    What  is  the  effect  of  dropping  iron-filings  into 
the  glass  vessel  1 

402.  In  what  way  was  Donny  able  to  raise  the  temper- 
ature of  water  to  135°  C.  without  ebullition  1 

403.  What  is  the  general  effect  of  salts  in  solution 
upon  the  boiling-point  of  water  ? 

404.  Give  examples  of  the  behavior  of  liquids  in  the 
spheroidal  state. 

405.  In  what  way  can  the  behavior  of  liquids  in  the 
spheroidal  state  be  explained,  and  what  experiment  of 
Boutigny  confirms  the  truth  of  this  explanation  ? 

406.  What  was  Faraday's  experiment  with  ether,  solid 
carbonic  acid,  and  mercury,  and  how  do  you  explain  the 
effects  which  he  observed  ? 

407.  Andrews  heated  liquid  carbonic-acid,  under  great 
pressure  in  a  closed  tube,  to  the  temperature  of  31°  C,,  or 
thereabouts.     What  phenomena  were  observed,  and  what 
inference  has  been  drawn  from  them  ? 

408.  What   is  sublimation,   and  what   are   instances 
of  it '] 

409.  What  is  the  effect  of  heating  a  strong  solution  of 
hydrochloric  acid  in  water  ?    of  heating  chalk  ? 

410.  Give  an  illustration  of  the  great  attraction  which 
some  gases  have  for  water. 

411.  What  six  gases  have  never  yet  been  condensed  by 
the  joint  effect  of  cold  and  pressure  1 

412.  Explain  what  is  meant  by  the  maximum  pressure 
of  a  vapor. 

413.  Show  how,  by  means  of  the  table  on  page  183, 
we  may  obtain  the  atmospheric  pressure  by  observations 
of  the  boiling-point  thermometer. 


LESS,  xxiv.]     ELEMENTARY  PHYSICS.  37 

414.  What  was  the  discovery  of  Gay  Lussac  in  refer- 
ence to  the  densities  of  gases  ?     Illustrate  it  by  the  case 
of  hydrogen  and  chlorine. 

415.  Give  a  recapitulation  of  the  effects  of  heat  which 
have  already  been  considered. 

416.  In  addition  to  the  effects  already  discussed,  enu- 
merate other  ways  in  which  heat  influences  bodies. 


LESSON  XXIV.  —  Conduction  and  Convection. 

417.  In  what  way  do  we  derive  our  heat  from  the  sun  1 

418.  Give  an  instance  of  the  mode  of  distribution  of 
heat  called  conduction.    How  does  it  differ  from  radiation  ? 

419.  What  example  does  this  branch  of  the  science  of 
heat  afford  of  the  provision  by  nature  for  the  welfare  of 
the  animal  creation  ? 

420.  For  what  two  purposes  may  a  bad  conductor  of 
heat  be  employed  ?     Give  an  instance  of  each. 

421.  Give  an  experiment  which  shows  the  difference 
between   the   conducting   power   of  two    different   sub- 
stances. 

422.  Why  is  it  that,  when  a  metal  bar  has  one  of  its 
extremities  heated  in  the  fire,  the  other  extremity  does 
not  ultimately  attain  the  same  temperature? 

423.  Give  Fourier's  definition  of  the  conductivity  of  a 
substance. 

424.  Suppose  we  have  two  bars  of  the  same  shape, 
size,  arid  conductivity,  but  unlike  in  material,  the  ends 
of  which  we  heat  by  a  spirit-lamp  to  the  same  extent ; 
and  let  the  surfaces  of  both  bars  be  covered  with  gilt. 
Now,  if  we  observe  the  temperatures  of  both  bars  at  equal 
distances  from  the  lamp,  we  shall  find  them  unequal  or 
equal  according  as  a  short  time  or  a  considerable  interval  has 
elapsed  since  the  lamp  was  first  applied.    Explain  this. 

425.  If  we  take  two  precisely  similar  pieces,  one  of 
'•oismuth  and  the  other  of  iron,  and,  coating  one  end  of 

each  with  white  wax,  place  the  other  end  in  a  hot  ves- 


38  QUESTIONS  ON  STEWART'S      [CHAP.  v. 

sel,  we  shall  find  that  the  wax  will  melt  first  on  the  bis- 
muth, although  iron  is  the  best  conductor.  Account  for 
this. 

426.  Explain  the  principle  of  Davy's  safety-lamp. 

427.  What  peculiarity  do  crystals  exhibit  in  their  con- 
ducting power,  and  how  was  this  shown  experimentally 
by  De  Senarmont  ? 

428.  How  may  the  bad  conducting  power  of  water  be 
shown  by  experiment  ? 

429.  Describe  the  process  called  convection.    How  may 
convection  currents  be  rendered  visible  ? 

430.  Give  an  account  of  convection  on  the  large  scale, 
as  exemplified  by  the  freezing  of  a  lake.     What  conse- 
quences would  ensue  if  water  had  no  point  of  maximum 
density  and  ice  were  heavier  than  water  ? 

431.  Upon  what  two  things  does  convection  depend  1 
How  is  this  illustrated  by  the  atmosphere  of  the  sun  1 

432.  Explain  the  trade-winds. 

433.  Explain  the  land  and  sea  breezes. 


LESSON  XXV.  —  Specific  and  Latent  Heat. 

434.  Define  the  unit  of  heat ;  and  also  the  specific  heat 
of  a  substance. 

435.  Explain  the  method  of  determining  the  specific 
heat  of  a  substance  called  the  "  method  of  mixtures "  by 
the  aid  of  a  numerical  example. 

436.  What  other  methods  of  estimating  specific  heat 
have  been  devised  1 

437.  In  general,  what  influence  have  temperature  and 
density  respectively  on  the  specific  heat  of  solids  ? 

438.  Generally  speaking,  do  substances  have  a  greater 
specific  heat  in  the  solid  or  in  the  liquid  state  ? 

439.  What  substance  has  generally  been  supposed  to 
have  the  greatest  specific  heat  1 

440.  What  two  kinds  of  specific  heat  may  be  distin- 
guished in  the  case  of  a  gas  ? 


LESS,  xxv.]       ELEMENTARY  PHYSICS.  39 

441.  What  results  did  Kegnault  obtain  respecting  the 
specific  heats  of  gases  1 

442.  What  law  did  Dulong  and  Petit  discover  with  re- 
spect to  the  specific  heats  and  atomic  weights  of  simple 
substances  ? 

443.  Under  what  circumstances  does  heat  become  la- 
tent ?     How  may  we  correctly  describe  the  condition  of 
water  at  0°  0.,  or  of  steam  at  100°  C.  ? 

444.  How  were  Black's  first  experiments  upon  latent 
heat  performed  ?     How  were  his  subsequent  experiments 
performed  so  as  to  measure  the  latent  heat  of  one  kilo- 
gramme of  water  ? 

445.  What  is  the  value,  in  the  metric  system,  of  the 
latent  heat  of  water  ]  of  the  latent  heat  of  steam  ? 

446.  What  important  parts  do  the  facts  that  water  has 
a  greater  latent  heat  than  any  other  substance,  and  that 
steam  has  a  greater  latent  heat  than  any  other  gas,  play  in 
the  economy  of  nature  ? 

447.  Viewing  heat  as  a  species  of  molecular  energy, 
what  twofold  office  does  it  discharge  ? 

448.  What  explanation  of  the  phenomona  connected 
with  latent  heat  is  furnished  by  the  doctrine  of  energy  ? 

449.  What  is  the  principle  of  all  freezing  mixtures 
and  processes  ? 

450.  Explain  the  method  of  using  the  wet  and  dry 
bulb  thermometers  for  estimating  the  hygrometric  state 
of  the  air. 

451.  How  did  Leslie  freeze  water  by  means  of  its  own 
evaporation  ? 

452.  Describe  Carre's  apparatus  for  artificially  pro- 
ducing ice  ? 

453.  How  did  Faraday  succeed  in  freezing  mercury  ? 

454.  Explain   the  fall  of   temperature  which  results 
from  mixing  snow  and  salt  together. 


40  QUESTIONS  ON  STEWART'S      [CHAP.  v. 

LESSON  XXVI.  —  On  the  Relation  between  Heat  and  Me- 
chanical Energy. 

455.  Give  instances  of  the  conversion  of  mechanical 
energy  into  heat.     In  what  case  is  this  conversion  unde- 
sirable, and  what  means  are  taken  to  avoid  it  ? 

456.  How  did  Joule  conduct  his  experiments  on  the 
relation  between  mechanical  energy  and  heat  1 

457.  What  is  the  mechanical  equivalent  of  heat  as  de- 
termined by  Joule  ? 

458.  What  was  the  nature  of  Mayer's  method  of  calcu- 
lating the  mechanical  equivalent  of  heat  ? 

459.  If  we  drop  a  weight  into  a  large  quantity  of  ful- 
minating powder,  the  result  is  the  generation  of  a  large 
amount  of  heat  ;  are  we  at  liberty  to  suppose  that  all  this 
heat  is  the  mechanical  equivalent  of  the  energy  of  the 
weight  1 

460.  What   question   similar  to   the   above    may  be 
asked  in  the  case  of  the  compression  of  a  gas,  and  what 
answer  to  it  is  supplied  by  Joule's  experiments  ? 

461.  Give  the  mechanical  explanation  of  the  fact  that 
a  gas  suddenly  expanded  becomes  cooled. 

462.  Explain,   with   the  aid  of  a  diagram,   how  the 
alternating  motion  of  the  piston  in  the  cylinder  of  a  steain- 
engiiie  is  produced  through  the  agency  of  steam. 

463.  What  are  the    chief   differences   between  high- 
pressure  and  low-pressure  engines  ? 

464.  What  is  the  general  law  for  the  conversion  of 
heat  into  mechanical  energy  ?     How  is  this  law  exempli- 
fied by  low-pressure  and  high-pressure  engines  respec- 
tively ? 

465.  State  Carnot's  analogy  between  the  mechanical 
capability  of  heat  and  that  of  water? 

466.  What  is  the  absolute  zero  of  temperature,  and  its 
value  on  the  centigrade  scale  1 

467.  Under  what  conditions  would  it  be  possible  to 
convert  all  the  heat  which  passes  through  a  heat-engine 
into  mechanical  effect  1 


LESS,  xxvi.]      ELEMENTARY  PHYSICS.  41 

468.  Suppose  that  the  higher  temperature  of  a  heat- 
engine  is  100°  C.,  and  the  lower  0°  C.  ;  what  proportion 
of  the  whole  heat  carried  through  the  engine  may  be  con- 
verted into  mechanical  effect  ? 

469.  What  is  the  general  rule  for  finding  how  much 
of  the  heat  carried  through  a  heat-engine  can  be  utilized  ? 

470.  What  difference  in  general  is  there  between  the 
theoretical  and  the  practical  limit  of  utilization  ? 

471.  What  three  kinds  of  heat-engines  are  in  extensive 
use  } 

472.  Give  a  sketch  of  the  history  of  heat-engines  prior 
to  the  time  of  Watt. 

473.  How  was  Watt's  attention  drawn  to  the  subject, 
and  what  were  the  three  great  improvements  which  he 
effected  ? 

474.  Explain  the  advantages   of  Watt's   arrangement 
for  the  condensation  of  the  steam. 

475.  Explain  the  principle  of  double  action  introduced 
by  Watt. 

476.  Explain  the  mode  of  expansive  working. 

477.  How  is  the  rate  at  which  an   engine  performs 
work  usually  expressed,  and  what  is  the  unit  employed 
for  this  purpose  called  1    What  is  its  numerical  value  in 
this  country  ] 

478.  On  reviewing  the  relations  of  heat  and  mechanical 
effect,  what  important  difference  in  their  mutual  converti- 
bility is  apparent  1 


42  QUESTIONS  ON  STEWART'S    [CHAP.  vi. 


CHAPTER  VI. 

EADIANT   ENERGY. 

LESSON  XXVII.  —  Preliminary. 

479.  What  is  the  velocity  with"  which  radiant  energy 
is  propagated  ? 

480.  Mention  phenomena  which  naturally  lead  to  a 
division  of  radiation  into  non-luminous  and  luminous,  or 
into  rays  of  dark  heat  and  rays  of  light. 

481.  Define  Optics. 

482.  What  are  the  two  hypotheses  respecting  the  na- 
ture of  light  ?     Why  has  the  new  hypothesis  the  better 
claim  to  be  regarded  as  true  ? 

483.  Define  the  following  terms  :  a  ray  of  light,  &  pen- 
cil of  rays,  divergent  pencil,  convergent  pencil,  pencil  of 
parallel  rays. 

484.  Into  what  two  distinct  classes  are  substances  di- 
vided with  reference  to  their  effect  upon  light  1 

485.  What  is  the  effect  of  allowing  light  to  fall  on  a 
very  thin  slice  of  an  opaque  substance  '?    What  does  this 
show  as  to  the  distinction  between  opaque  and  transpar- 
ent substances  ? 

486.  What  two  important  exceptions  are  there  to  the 
law  that  light  moves  in  straight  lines  1 

487.  Explain  (with  a  diagram)  how  Romer  was  able  to 
determine  the  velocity  of  light  from  the  eclipses  of  Ju- 
piter's satellites. 

488.  Explain  Fizeau's  method  of  measuring  the  velo- 
city of  light. 

489.  Prove  that  the  quantity  of  light  which  a  surface 
receives  from  any  source  will  vary  inversely  as  the  square 
of  its  distance  from  the  source. 


LESS,  xxvui.]  ELEMENTARY  PHYSICS.  43 

490.  Show  that  the  intensity  of  the  illumination  of  a 
plate  or  screen  is  proportional  to  the  cross-section  which 
it  presents  to  the  direction  of  radiation. 

What  familiar  facts  respecting  the  power  of  the  sun's 
rays  are  accounted  for  by  this  law  ? 

491.  Prove  that  the  intrinsic  brightness  of  a  luminous 
body  does  not  vary  with  its  distance  ;  meaning  by  bright- 
ness the  light  that  would  reach  the  eye  by  looking  at  the 
body  through  a  long  narrow  tube,  and  supposing  the  tube 
to  be  always  so  narrow,  and  the  source  of  light  always  so 
large,  that  in  looking  through  the  tube  we  should  see 
nothing  else  but  this  light. 

Consider  also  the  case  in  which  the  luminous  body  is 
so  distant  as  to  appear  simply  a  luminous  point  like  a 
star. 

492.  Explain  Bunsen's  photometer,  and  the  method  of 
measuring  the  intensity  of  light  by  means  of  it. 

493.  Suppose  that  one  light  causes  the  grease  spot  to 
vanish  in  a  Bunsen's  photometer  when  placed  at  the  dis- 
tance of  one  foot  in  front  of  the  screen,  and  another  light 
when  placed  at  the  distance  of  two  feet  ;  what  is  the  rela- 
tive luminosity  of  the  two  lights  ? 

494.  Explain  the  distinction  between  the  illuminating 
power  of  a  source  of  light  and  the  inherent  brightness  or 
quality  of  the  light. 

LESSON  XXVIII.  —Reflection  of  Light. 

495.  State  the  law  of  reflection. 

496.  How  may  the  truth  of  this  law  be  rendered  visi- 
ble to  the  eye  ? 

497.  What  is  meant  by  a  virtual  image  1 

498.  Prove  by  the  aid  of  a  figure  that  the  image  of  a 
luminous  point  lies  as  far  behind  the  reflecting  surface  as 
the  luminous  point  itself  lies  before  it. 

499.  Explain  by  a  figure  the  mode  of  determining  the 
positions  of  the  various  points  in  the  image  of  a  luminous 
body  which  is  in  front  of  a  plane  mirror. 


44  QUESTIONS  ON  STEWART'S    [CHAP.  vi. 

500.  What  peculiar  inversion  is  there  in  the  reflection 
of  the  human  figure  in  a  vertical  mirror  ?     Also,  in  the 
reflection  of  letters  written  from  left  to  right  on  a  wall  in 
front  of  the  mirror  ? 

501.  When  a  ray  of  light  strikes  a  curved  surface,  how 
may  the  direction  of  the  reflected  ray  be  found  ? 

502.  Suppose  a  pencil  of  parallel  rays  strikes  a  con- 
cave spherical  mirror  ;  prove  that  the  focus  of  the  rays  is 
the  point  half-way  between  the  centre  of  the  mirror  and 
the  middle  point  of  its  surface.    What  is  this  focus  called  ? 
(See  page  229.) 

503.  Explain    how  a  concave    spherical    mirror  pro- 
duces a  circular  image   of  the  sun.     Will  this  image  be 
real  or  virtual  ? 

504.  Show  by  a  diagram  that  the  focus  of  divergent 
rays  proceeding  from  a  point  near  a  concave  spherical 
mirror  lies  between  the  principal  focus  and  the  centre  of 
the  mirror. 

505.  What  are  conjugate  foci,  and  what  is  meant  by 
saying  that  conjugate  foci  are  interchangeable  ? 

506.  Prove  the  formula  which  gives  the  relation  be- 
tween the  conjugate  foci  of  a  concave  spherical  mirror. 

507.  Show  that  a  virtual  image  will  be  produced  if 
the  luminous  point  be  nearer  the  concave  mirror  than  the 
principal  focus. 

508.  Examine  the  five  different  cases  to  wrhich  the 

112 

general  formula  -  -f-  —  =  -  is  applicable. 

509.  The  images  produced  by  concave  mirrors  are  in 
their  nature  either  real  or  virtual,  in  their  position  rela- 
tively to  the  object  either  erect  or  inverted,  in  size  com- 
pared with  the  object  either  magnified  or  diminished.    Ex- 
amine as  regards  these  particulars  the  image  of  an  object 
such  as  a  straight  line  placed  beyond  the  centre  of  the  mir- 
ror. 

510.  Examine  as  regards  the  above  particulars  the  im- 
age of  an  object  placed  between  the  principal  focus  and 
the  concave  mirror. 


LESS,  xxix.]     ELEMENTARY  PHYSICS.  45 

511.  An  object  is  placed  immediately  in  front  of  a  con- 
cave mirror,  and  then  gradually  removed  to  a  great  dis- 
tance along  the  axis  of  the  mirror  ;  trace  the  changes  in 
the  distance  of  the  image  from  the  mirror,  and  also  in  the 
nature,  position,  and  size  of  the  image. 

512.  Explain,  with  a  diagram,  the  effect  produced  by 
a  parabolic  mirror  upon  a  pencil  of  parallel  rays  incident 
on  its  surface.     What  advantage  do  parabolic  mirrors  pos- 
sess as  compared  with  spherical  mirrors  ?     On  the  other 
hand,  what  disadvantage  I 

513.  What  is  the  nature  of  the  images  produced  by 
convex  spherical  mirrors  ? 


LESSON  XXIX.  —  Refraction  of  Light. 

514.  Illustrate,  by  a  diagram,  the  refraction  of  light  by 
the  surface  of  a  transparent  medium  like  glass,  and  state 
the  law  of  refraction  in  general  terms. 

515.  Instead  of  introducing  the  sines  of  the  angles  of 
incidence  and  refraction  in  the  statement  of  the  law  of 
refraction,  how  can  the  law  be  expressed  in  purely  geomet- 
rical language  ? 

516.  Explain  the  case  in  which  the  ray  of  light,  instead 
of  passing  from  vacuo  into  a  transparent  medium,  passes 
out  from  the  medium  into  vacuo. 

517.  Explain  the  case  in  which  a  ray  of  light  strikes 
the  surface  of  a  medium  at  right  angles. 

518.  Explain  how  the  truth  of  the  laws  of  refraction 
may  be  illustrated  experimentally. 

519.  Explain  total  internal  reflection.    What  is  the  criti- 
cal angle  of  a  medium  ? 

520.  Explain  the  mirage. 

521.  If  nv  n2  be  the  absolute  indices  of  refraction  of  two 
media  respectively,  and  ri  be  the  relative  index  of  refrac- 
tion for  the  two  media,  prove  that  n  =  — . 

HI 

522.  Trace,  by  means  of  a  figure,  the  path  of  a  ray  of 


46  QUESTIONS  ON  STEWART'S    [CHAP.  vi. 

light  through  a  glass  prism.     What  is  the  angle  of  devi- 
ation ? 

523.  What  is  the  condition  of  minimum  deviation  in  a 
prism  1 

524.  What  is  the  condition  of  total  internal  reflection  in 
a  prism  1 

525.  Why  can  we  not  employ  for  ordinary  optical  pur- 
poses a  glass  prism,  of  which  the  angle  is  greater  than 
84°? 


LESSON  XXX.  —  Lenses  and  other  Optical    Instruments. 

526.  Describe  the  shapes  given  to  the  lenses  in  common 
use,  and  give  the  names  of  the  lenses 

527.  From  the  action  of  a  prism  on  a  ray  of  light  de- 
rive a  rule  for  determining  whether  a  lens  is  converg- 
ing or  diverging ;  and  apply  this  rule  to  the  six  lenses 
mentioned  in  the  book. 

528.  Discuss  the  formula  which  expresses  the  relation 
between  the  conjugate  foci  of  a  double  convex  lens,  exam- 
ining the  different  cases  which  arise  as  the  luminous  point 
is  supposed  to  move  along  the  axis  of  the  lens  from  an  in- 
finite distance  up  to  the  lens.    The  formula  is  -  +  -j-  =  ~, 

in  which  p,  p   are  the  conjugate  foci,  and  /  the  principal 
focus  of  the  lens. 

529.  Show,  by  a  figure,  how  to  find  the  position  and 
size  of  the  image  of  a  luminous  body  formed  by  a  double 
convex  lens,  the  luminous  body  being  supposed  to  be 
farther  from  the  lens  than  the  principal  focus.     What  is 
the  law  which  determines  the  size  of  the  image  compared 
with  that  of  the  object  1 

530.  Show  that  if  a  luminous  body  be  placed  between 
a  double  convex  lens  and  its  principal  focus,  the  image 
will  be  virtual,  erect,  and  magnified. 

531.  Describe  the  two  sets  of  appearances  which  may 
be  seen  in  looking  through  a  double  convex  lens,  and  state 
the  conditions  under  which  they  are  produced. 


LESS,  xxxi.]     ELEMENTARY  PHYSICS.  47 

532.  Describe  the  camera  dbscura  and  its  use. 

533.  Describe  the  eye,  regarded  as  an  optical  instru- 
ment 

534.  What  power  of  adjustment  does  the  eye  possess, 
and  under  what  circumstances  is  this  power  called  into 
action  ? 

535.  When  is  a  person  said  to  be  short-sighted,  and 
when  long-sighted  ?   What  are  the  remedies  for  these  de- 
fects, respectively  ? 

536.  What  is  the  principle  of  the  simple  microscope  ? 
What  condition  must  be  answered  in  order  that  the  virtual 
image  which  is  formed  may  be  distinct  1 

537.  What  are  the  essential  parts  of  a  telescope  ?    Ex- 
plain by  a  figure  how  a  telescope  forms  a  virtual  and  mag- 
nified image  of  a  distant  object. 

538.  What  is  the  optical  difference  between  the  simple 
microscope  and  the  telescope  ? 


LESSON  XXXI.  —  Dispersion  of  Light  by  the  Prism. 

539.  What  great  discovery  did  Newton  make  as  to  the 
nature  of  white  light  ? 

540.  Explain  the  dispersion  of  light  by  a  prism.    Give 
the  seven  principal  colors  of  the  spectrum  in  the  order  of 
refrangibility. 

541.  Why  is  it  of  great  importance,  in  experiments 
upon  the  decomposition  of  light  by  prisms,  to  make  use 
of  a  very  narrow  slit  1 

542.  What  is  the  method,  employed  in  the  spectroscope, 
of  multiplying  the  dispersions  of  rays  of  different  refran- 
gibilities  1     Describe  briefly  the  spectroscope  of  Gassiot. 

543.  Explain  an  optical  method  of  recombining  the 
various  constituents  of  white  light. 

544.  Explain  a  mechanical  method  of  combining  the 
various  colors  of  the  spectrum  so  as  to  form  white  light. 


48  QUESTIONS  ON  STEWART'S    [CHAP.  vi. 

LESSON  XXXII.  —  Thermo-Pile. 

545.  In  what  way  can  we  compare  together  the  in- 
tensity of  a  ray  of  light  and  a  ray  of  dark  heat,  and  ob- 
tain a  true  measure  of  the  energy  of  these  rays,  provided 
instruments  of  sufficient  delicacy  be  employed  ? 

546.  Explain  the  principle  of  the  thermo-pile  discov- 
ered by  Seebeck. 

547.  In  order  to  obtain  by  the  use  of  thermo-electricity 
a  very  delicate  instrument  wherewith  to  measure  radiant 
heat,  what  three  objects  must  be  accomplished  ? 

548.  How  may  a  strong  thermo-electric  current  be 
produced  ?     Illustrate  with  a  figure. 

54'9.  Describe  Thomson's  galvanometer  and  its  action, 
explaining  in  particular  how  the  magnetic  force  is  over- 
come, and  how  by  optical  arrangements  any  small  motion 
of  the  needle  is  very  much  magnified. 

550.  Explain  the  construction  of  the  thermo-pile. 

551.  Suppose  that  when  a  source  of  radiant  heat  is 
placed  before  the  pile,  the  luminous  slit  is  made  to  move 
on  the  screen  through  twenty  divisions  of  the  scale  ;  sup- 
pose, again,  that  when  a  different  source  of  heat  is  pre- 
sented to  the  pile,  the  index  moves  over  forty  divisions  ; 
what  is  the  relation  between  the  heating  effects  of  the  two 
sources  ?    What  is  the  general  law  ? 

552.  Explain  how,  by  the  joint  aid  of  the  spectroscope 
and  the  pile,  we  are  enabled  to  analyze  a  beam  of  sun- 
light so  as  to  estimate  the  heating  effects  of  all  the  differ- 
ent rays  in  the  solar  spectrum.    What  weak  point  is  there 
in  this  method  1 

553.  What  source  of  heat  did  Leslie  employ  in  his  ex- 
periments on  dark  heat,  and  what  was  one  of  the  results 
at  which  he  arrived  ? 

554.  What  are  some  of  the  facts  established  by  Melloni 
with  the  aid  of  the  pile  ?     What  is  diathermancy  ? 

555.  Explain,  by  the  aid  of  a  figure,  the  manner  in 
which  Melloni  performed  his  experiment  to  prove  that 
dark  heat  is  capable  of  refraction. 


LESS.  XXXIIL]    ELEMENTARY  PHYSICS.  49 

556.  What  two  facts  in  reference  to  dark  heat  were  es- 
tablished by  Forbes  ? 

557.  Give  a  sketch  of  the  manner  in  which  we  may 
explore  experimentally  the  heat  spectrum  produced  by  a 
heated  strip  of  coal,  for  example. 

558.  Suppose  we  begin  by  heating  a  strip  of  carbon  to 
a  heat  below  redness,  and  producing  a  spectrum  of  the 
radiation  from  the  carbon  by  means  of  a  rock-salt  prism  ; 
trace  the  changes  in  this  spectrum  as  the  temperature  of 
the  carbon  is  gradually  raised  to  a  very  high  point. 

559.  What  is  the  position  upon  the  spectrum  of  actinic 
rays  as  they  are  called,  and  what  power  do  they  possess  ? 

560.  Give  a  graphical  representation  of  the  sun's  visi- 
ble spectrum,  locating  the  primary  colors,  and  drawing 
the  curve  of  intensity  of  light. 

561.  Give  a  graphical  representation  of  the  entire  solar 
spectrum,  and  draw  the  curve  of  intensity  of  heat. 

562.  Where  is  the  maximum  luminous  effect  in  the 
solar  spectrum,  and  where  the  maximum  heating  effect  ? 

563.  Explain  the  statement  "  the  spectrum  of  carbon  is  a 
continuous  one"     In  general,  what  bodies  give  continuous 
spectra  1 

564.  In  what  respect  do  the  spectra  of  gases   differ 
from  those  of  solid  bodies  1    What  is  the  spectrum  of 
ignited  sodium  vapor  1  of  thallium  1 

565.  What  chromatic  phenomena  will  be  observed  if 
we  ignite  a  piece  of  metallic  sodium  in  a  dark  room  1 

566.  In  what  way  has  electricity  been  found  service- 
able in  spectrum-analysis  ? 


LESSON  XXXIII.  —  Radiation  and  Absorption. 

567.  What  great  and  striking  difference  between  the 
spectra  of  solids  and  those  of  incandescent  gases  was  made 
known  in  the  last  Lesson  ? 

568.  Explain  how  it  may  be  shown  by  experiment 
that  at  comparatively  low  temperatures,  say  100°  C.,  a 

3  D 


50  QUESTIONS  ON  STEWART'S    [CHAP.  vi. 

lamp-black  surface,  or  one  of  glass  or  white  paper,  radiates 
much  more  than  a  surface  of  polished  silver. 

569.  Explain  how  the  relative  absorbing  powers  at 
100°   C.  of  the  substances  mentioned  in  the  preceding 
question  may  be  determined  experimentally. 

570.  On  comparing  two   tables,  one   containing  the 
radiating  and  the  other  the  absorbing  powers  of  a  series 
of  substances,  what  general  law  comes  to  view  ? 

571.  In  what  important  respect  do  surfaces  differ  as 
regards  their  absorbing  powers  for  different  rays  ?    What 
instances  of  this  can  you  give  ? 

572.  Describe  three  experiments  illustrative   of  the 
radiation  from  bodies  of  high  temperature.     What  gen- 
eral relation  between  absorption  and  radiation  do  these 
experiments  tend  to  establish  ? 

573.  What  is  found  to  be  the  behavior  of  transparent 
colorless  glass  as  regards  absorption  and  radiation  ;  also, 
of  a  film  or  stratum  of  air  ? 

574.  Give  a  generalization  of  the  conclusions  to  be 
drawn  from  the  preceding  experiments. 

575.  Explain  what  is  meant  by  selective  or  partial  ab- 
sorption by  describing  the  behavior  of  white  paper,  and 
also  of  the  glass  bulb  of  a  thermometer,  at  different  tem- 
peratures. 

576.  What  is  it  that  makes  the  leaves  of  plants  appear 
green  ?     In  general,  what  is  the  physical  cause  of  color  ? 

577.  What  familiar  illustration  of  selective  absorption 
is  afforded  by  colored  glasses  ? 

578.  Describe  an  experiment  in  proof  of  the  law  that 
bodies  when  cold  absorb  the  same  kind  of  rays  that  they  give 
out  when  hot. 

579.  Describe   another    experiment  in  proof  of  the 
above  law. 

580.  Describe  a  third  experiment  in  proof  of  the  same 
law. 

581.  Suppose  that  we  introduce  into  a  chamber,  kept 
uniformly  at  a  white  heat,  transparent  glass,  polished  plat- 
inum, coal,  and  black  and  white  porcelain  ;    arid  that, 


LESS,  xxxm.]    ELEMENTARY  PHYSICS.  51 

after  leaving  them  until  they  have  acquired  the  tem- 
perature of  the  walls  of  the  chamber,  as  a  first  experiment 
we  simply  examine  them  through  a  small  hole  ;  finally, 
suppose  that,  as  a  second  experiment,  we  hastily  withdraw 
the  substances,  and,  without  allowing  them  time  to  cool, 
examine  them  in  the  dark  ;  —  what  will  be  the  appear- 
ances presented,  in  the  two  experiments,  and  how  may 
they  be  reconciled  with  one  another  1 

582.  If  we  introduce  red  and  green  glass  into  a  white- 
hot  chamber,  and  then  view  them  through  a  small  open- 
ing, they  will  appear  to  have  entirely  lost  their  color. 
Explain  this. 

583.  Give  the  grounds  upon  which  black  bodies  have 
been  selected  as  the  standard  or  typical  radiators. 

584.  What  simple  method  may  be  employed  to  ascer- 
tain whether  or  not  one  body  is  hotter  or  colder  than 
another  ? 

585.  Explain  how,  by  the  aid  of  the  spectroscope,  we 
may  learn  the  chemical  nature  of  a  substance. 

586.  Show  how  spectrum  analysis  has  demonstrated 
that  there  are  present  in  the  sun,  in  the  state  of  vapor, 
various  substances  well  known  on  the  earth,  as  sodium, 
iron,  zinc,  magnesium,  etc. 

587.  What  conclusion  may  we  draw  from  the  results 
of  Professor  Tyndall's  investigations  into  the  absorption 
of  various  gases  for  dark  heat  ? 

588.  Explain  the  part  which  the  aqueous  vapor  of  the 
atmosphere  plays  in  relation  to  the  heating  effect  of  the. 
sun  upon  the  earth's  surface. 

589.  Show  how  the  laws  of  radiation  explain  the  depo- 
sition of  dew. 

590.  Give  some  examples  of  the  phenomenon  called 
phosphorescence.    Also  give  an  instance  of  the  similar  phe- 
nomenon known  as  fluorescence. 

591.  What  is  Professor  Stokes's   explanation  of  the 
phenomena    of    phosphorescence   and    of    fluorescence  ] 
What  is  really  the  only  difference  between  the  two  phe- 
nomena ? 


52  QUESTIONS  ON  STEWART'S    [CHAP.  vi. 

LESSON  XXXIV.  —  On  the  Nature  of  Radiant  Energy. 

592.  What  two  hypotheses  regarding  the  nature  of 
light  were  propounded  by  Newton  and  Huyghens,  respec- 
tively ? 

593.  What  crucial  test  between  these  two  hypotheses 
has  been  found  ? 

594.  What  striking  analogy  is  there  between  light  and 
sound  which  leads  to  the  belief  that  light  must  be  a  mo- 
tion similar  to  sound,  that  is  to  say,  undulatory  ? 

595.  On  the  undulatory  theory,  how  does  the  eye  dis- 
tinguish between  rays  of  different  wave-lengths  ?     What 
analogy  is  there  in  this  particular  between  light  and 
sound  1 

596.  Illustrate  what  is  meant  by  the  front  of  a  wave, 
and  give  a  general  definition  of  the  same. 

597.  In  what  direction,  relatively  to  its  front,  does  a 
wave  always  proceed  1 

598.  Deduce  the  law  of  reflection  from  the  undulatory 
theory  of  light. 

599.  Deduce  the  law  of  refraction  from  the  undulatory 
theory  of  light. 

600.  In  the  undulatory  theory,  what  does  the  index  of 
refraction  of  a  substance  represent  1 

601.  What  reason  may  be  assigned  why  the  velocity 
of  light   should   be   less  in  glass,  for  example,  than  in 
vacuo  1 

602.  What  analogy  serves  to  aid  the  mind  in  perceiving 
why  reflection  and  refraction  accompany  each  other  when 
light  falls  on  a  polished  glass  surface,  for  example  ? 

603.  What  well-known  facts  respecting  shadows  might 
lead  us  to  imagine  that  light  differs  from  sound  in  a  fun- 
damental respect  1    What  is  the  cause  of  the  difference 
between  sound-shadows  and  light-shadows  1 

604.  Give  instances  of  the  manner  in  which  the  beauti- 
fully colored  appearances  due  to  the  interference  of  light 
may  be  produced.     What  is  the  general  explanation  of 
these  appearances  according  to  the  undulatory  theory  1 


LESS,  xxxv.]      ELEMENTARY  PHYSICS.  53 

605.  Explain  Newton's  rings. 

606.  Explain  the  colors  of  thin  plates,  such  as  those 
of  a  soap-bubble. 

607.  State   an  apparent  objection  to  the  undulatory 
theory,  derived  from  the  laws  of  energy,  and  show  how 
this  objection  is  entirely  removed. 

608.  Explain  why  it  is,  that,  when  a  sounding  body  is 
approaching  the  ear,  its  note  is  rendered  more  acute,  while 
if  it  be  receding  from  the   ear,  its  note  becomes  more 
grave. 

609.  Show  how  Mr.  Huggins  has  been  able  to  make 
out  the  proper  motions  of  several  stars  in  a  direction  to 
and  from  the  eye. 


LESSON  XXXV.  —  Polarization  of  Light.     Connection  be- 
tween Radiant  Energy  and  the  other  Forms  of  Energy. 

610.  What  two  kinds  of  wave-motion  are  met  with  in 
nature  ? 

611.  Which  kind  of  vibrations  is  capable  of  assuming 
a  particular  side  or  direction,  and  how  may  this  fact  be 
illustrated  ? 

612.  What  is  the  meaning  of  the  term  polarization  ? 

613.  Give  an  illustration  to  show  how  a  mixture  of 
vertical  and  horizontal  waves  may  be  sifted,  so  to  speak, 
and  deprived  of  the  vertical  components  of  the  waves, 
or  of  the  horizontal  components,  or  of  both. 

614.  Describe  the  action  of  tourmaline  upon  light. 

615.  What  is  the  only  possible  explanation  of  the  phe- 
nomena which  are  observed  ?     To  whom  are  we  indebted 
for  this  explanation  ] 

616.  How  is  polarization   by  reflection  effected,  and 
what  is  meant  by  saying  that  the  light  is  then  "  polarized 
in  the  plane  of  reflection "  1 

617.  Show  that  an  ordinary  ray  of  light  may  be  made 
to  disappear  entirely  by  two  reflections. 

618.  Explain  the  double  refraction  of  light  by  a  crystal 


54  QUESTIONS  ON  STEWART'S      [CHAP.  vi. 

of  Iceland  spar.     What  is  the  appearance  of  a  small  body 
as  seen  through  a  piece  of  Iceland  spar  ? 

619.  Explain  the  general  connection  between  radiant 
energy,  mechanical  energy,  and  the  energy  of  absorbed 
heat. 


LESS,  xxxvi.]     ELEMENTARY  PHYSICS.  55 

CHAPTER  VII. 

ELECTRICAL  SEPARATION. 

LESSON  XXXVI.  —  Development  of  Electricity. 

620.  Mention  two   leading  facts  in  the  early  history 
of  electricity.     From  what  is  the  word  derived  ? 

621.  What  marked  difference  exists  between  metal  and 
glass  as  regards  their  power  to  conduct  electricity  ?    By 
what  terms  do  we  express  this  difference  ? 

622.  Give  the  tables  of  the  most  important  conductors 
and  insulators.     What  is  the  character  of  the  transition 
from  the  one  class  of  bodies  to  the  other  ? 

623.  Why  is  it  very  desirable  to  make  all  experiments 
on  electricity  in  a  dry  atmosphere  1 

624.  Show  by  an  experiment  that  there  are  two  kinds 
of  electricity,  and  give  their  names.     When  do  electrified 
bodies   attract  each  other,  and  when  do  they  repel  each 
other  ? 

625.  Explain  the  hypothesis  of  two  fluids. 

626.  Give  a  table  of  twelve  common  substances  in  the 
order  of  their  relative  capacity  for  positive  electrification. 

627.  Mention  other  modes  of  developing  electrical  sep- 
aration besides  friction. 

628.  What  appears  to  be  an  essential  condition  for 
the  production  of  electricity  by  the  mutual  action  of  two 
bodies  1 

629.  What  general  connection  is  there  between  electri- 
cal separation  and  energy  or  mechanical  work  ? 

630.  Describe  the  electrical  properties  of  tourmaline. 
What  species  of  energy  is  spent  in  this  case  to  produce  the 
electrical  separation  1 


56  QUESTIONS  ON  STEWART'S   [CHAP.  vn. 

LESSON  XXXVII.  —  Measurement  of  Electricity. 

631.  Show  how  an  electrical  charge  upon  a  metallic 
body  can  be  subdivided. 

632.  Describe  Coulomb's  torsion-balance,  and  the  experi- 
ments which  demonstrate  the  law  of  electrical  action  be- 
tween two  bodies,  so  far  as  it  depends  on  the  distance  of 
the  bodies  from  each  other. 

633.  Explain  how,  by  means  of  Coulomb's  torsion-bal- 
ance, we  may  prove  the  law  of  action  between  two  electri- 
fied bodies,  so   far  as  it   depends  on  the  quantities  of 
electricity  upon  the  bodies. 

634.  What  is   a   convenient  unit  of   electrical  force  1 
Find  in  terms  of  this  unit  the  force  exercised  by  6  units 
of  positive  upon  4  units  of  negative  electricity  at  the  dis- 
tance 3. 

635.  Show,  by  an  experiment,  that  electricity  mani- 
fests itself  only   on  the  surface  of  bodies,   and  give  the 
explanation  of  this  fact. 

636.  In  certain  countries  electrical  manifestations  are 
often  produced  by  combing  the  hair,  rubbing  a  silk  dress, 
etc.,  while  they  are  not  observed  in  other  parts  of  the 
world.     How  do  you  account  for  this  ? 

637.  In  what  way  does  a  charge  of  electricity  distribute 
itself  on  a    sphere  1   on    a  pointed   conductor  ?     What 
accounts  for  the  distribution  in  each  case  ? 

638.  Define  the  term  electric  density.    Describe  a  body 
such  that  the  electric  density  will  be  much  greater  at  some 
parts  than  at  others. 

639.  Show  how  the  relative  distribution  of  electricity 
over  the  surface  of  a  body  may  be  ascertained  by  means 
of  the  proof-plane. 


LESSON  XXXVIII. — Ekctrical  Induction. 

640.  What  will  happen  if  we  bring  near  together  two 
insulated  conductors,  one  charged  with  electricity,  and  the 
other  not  charged  ?  What  is  this  kind  of  action  called? 


LESS,  xxxix.]     ELEMENTARY  PHYSICS.  57 

641.  Suppose  tfcat  the  neutral  conductor  in  the  pre- 
ceding question  be  divided  into  two  parts,  what  will  be  the 
electrical  condition  of  each  part  1     How  may  this  fact  be 
proved  by  experiment  1 

642.  Suppose  that  we  slowly  bring  a  conductor,  charged 
with  electricity,  towards  another  conductor  not  charged, 
until  they  are  very  near  each   other ;  explain  the  phe- 
nomena which  will  take  place.  • 

643.  How  may  it  be  rendered  evident  that  the  induc- 
tive effect  of  electricity  depends  on  the  distance  between 
the  two  conductors  ? 

644.  What  new  light  does  electrical  induction  throw 
upon  the  fact  that  electricity  only  shows  itself  at  the  sur- 
faces of  bodies  1 

645.  What  important  fact  was  discovered  by  Faraday 
in  his  researches  upon  electrical  induction  ?    What  is  the 
inductive  capacity  of  a  substance  2 


LESSON  XXXIX.  —  Electrical  Machines,  etc. 

646.  Of  what  two  parts  is  every  electrical  machine 
composed  ? 

647.  Describe  the  plate  electrical  machine,  and  explain 
its  action. 

648.  Describe  the  simple  experiments  with  an  electri- 
caL machine  which  may  be  performed, — 

1.  by  holding  the  finger  near  the  charged  conductor  ; 

2.  by  placing  an  individual  on  an  insulating  stool  ; 
and  give  the  explanation   of  the  experiments  according 
to  the  two-fluid  theory. 

649.  Describe  the  electrophorus,  and  explain  its  action. 

650.  Describe  the  gold-leaf  electroscope,  and  explain  how 
it  enables  us  not  only  to  detect  the  presence  of  electricity, 
but  also  to  determine  whether  the  electricity  is  positive 
or  negative. 

651.  What  difference  between  an  electroscope  and  an 
electrometer  is  indicated   by  the   derivation   of  the  two 
words  themselves  ? 


58  QUESTIONS  ON  STEWART'S    [CHAP.  vii. 

652.  Explain    the    method    of    measuring    electrical 
charges  employed  by  Sir  W.  Thomson  in  his  electrom- 
eters. 

653.  Explain  the  accumulation  of  electricity  by   con- 
densers.    If  the  condensing  plates  are  separated,  the  pith- 
balls  attached  to  them  will  diverge  ;  explain  this. 

654.  Describe  the  Ley  den  jar.     Show  how  it  may  be 
charged  and  discharged,  and  explain  its  mode  of  action. 

655.  If  a  Leyden  jar  be  allowed  to  stand  for  a  short 
time  after  being  discharged,  it  is  found  that  it  has  a  small 
residual  charge  left  in  it ;  what  is  the  probable  explana- 
tion of  this  ? 

656.  What  is  an  electric  battery,  and  how  formed  from 
its  component  parts  ? 

657.  What  knowledge  of  the  nature  of  the  electric 
spark  has  been  obtained  by  viewing  it  through  the  spec- 
troscope, and  what  use  has  been  made  of  this  knowledge  ? 

658.  What  transmutation  of  energy  do  we  have  in  the 
electric  spark  ? 

659.  Investigate  the  relation  between  the  charge  of  a 
Leyden  jar  and  the  amount  of  heat  produced  by  discharg- 
ing the  jar,  showing  that  the  whole  heating  effect  will  be 
proportional  to  the  square  of  the  quantity  of  electricity 
divided  by  the  surface  of  the  jar. 

660.  How  can  it  be  shown,  experimentally,  that  the 
duration  of  the  electric  spark  is  exceedingly  short  ? 

661.  How  has  Sir  C.  Wheatstone  succeeded  in  measur- 
ing the  duration  of  the  electric  spark  ?     What  was  the  re- 
sult of  his  experiments  ]     What  did  he  also  find  to  be 
the  velocity  of  electricity  1 

662.  Who  first  proved  that  lightning  is  only  a  manifes- 
tation of  electricity  on  a  large  scale,  and  in  what  way  did 
he  prove  that  this  is  the  case  1     What  advantage  has  been 
taken  of  this  knowledge  ? 

663.  To  what  are  the  following  phenomena  due  ?  — 

1.  The  light  which  constitutes  the  electric  flash. 

2.  The  noise  which  accompanies  the  same. 

3.  Its  destructive  effect  in  rending  substances. 


LESS,  xxxix.]     ELEMENTARY  PHYSICS.  59 

664.  If  we  bring  a  hollow  insulated  brass  ball  near  an 
electric  machine  in  action,  we  shall  get  a  spark,  but  it 
will  be  very  feeble.     If,  however,  we  touch  with  our  fin- 
ger that  part  of  the  conductor  which  is  farthest  from  the 
machine,  or  make  a  connection  between  this  conductor 
and  the  ground,  the  spark  from  the  machine  will  be  much 
more  intense.     Explain  this. 

665.  Show  how  to  obtain  from  an  insulated  conductor, 
near  an  electric  machine  in  action,  — 

1.  a  series  of  sparks  or  shocks  ; 

2.  a  continuous  rush  of  electricity. 

666.  Explain  the  efficacy  of  lightning-conductors. 

667.  Discuss  briefly  the  connection  between  electrical 
separation  and  the  other  forms  of  energy. 


60  QUESTIONS  ON  STEWART'S  [CHAP.  vm. 

CHAPTER    VIII. 

ELECTRICITY  IN  MOTION. 

LESSON  XL.  —  Magnetism. 

668.  What  is  the  origin  of  the  term  Magnet  ? 

669.  Describe   some   of  the  properties   of  a  magnet. 
What  are  its  poles,  and  how  are  they  distinguished  from 
each  other? 

670.  What  is  the  difference  between  magnetic  and  dia- 
magnetic  bodies  ?    Enumerate  the  most  important  bodies 
of  each  class.     In  what  respect  does  iron  stand  alone  ? 

671.  Explain  the  behavior  of  magnetic  and  diamagnetic 
bodies  when  suspended  midway  between  the  two  poles  of 
a  powerful  magnet. 

672.  Explain  the  behavior  of  magnetic  and  diamag- 
netic bodies  when  suspended  between  the  poles  of  a  mag- 
net in  a  magnetic  liquid  instead  of  in  air. 

673.  What  is  the  law  of  the  mutual  action  of  magnetic 
poles  ? 

674.  State  the  quantitative  law  of  force  in  magnetic 
attractions  and  repulsions.     By  whom  was  this  law  dis- 
covered ? 

675.  Prove  that,  in  consequence  of  this  law,  if  we  sus- 
pend a  small  magnet  by  a  thread  and  cause  it  to  approach 
the  pole  of  a  powerful  magnet,  the  small  magnet  will 
exhibit  a  tendency  to  rush  bodily  to  the  large  magnet ; 
and  find  the  measure  of  this  tendency. 

676.  Explain  magnetic  induction. 

677.  Describe  the  effect  of   breaking   a  magnet,  and 
give  a  theory  of  the  distribution  of  the  magnetic  fluids 
which  will  explain  the  properties  of  magnets,  both  when 
entire  and  when  broken. 

678.  What  difference  is  there  between  soft  iron  and 


LESS.  XLI.]        ELEMENTARY  PHYSICS.  61 

hard  steel  as  regards  susceptibility  to  magnetism  1  Ex- 
plain one  mode  of  magnetizing  a  steel  bar.  What  is  the 
effect  of  heat  on  magnets  1 

679.  If  we  were  to  suspend  a  magnetic  needle  in  such 
a  manner  that  it  was  perfectly  free  to  move  in  any  direc- 
tion, how  would  it  place  itself  ? 

680.  What  are  magnetic  meridians  ? 

681.  What  facts  are  stated  as  showing  that  a  magnetic 
needle  will  not  everywhere  and  always  point  as  it  does  in 
Great  Britain  at  the  present  moment  1 

682.  At  what  places  on  the  earth's  surface  is  a  mag- 
netic needle  of  no  use  to  the  mariner,  and  why  1 

683.  Explain  why  the  effect  of  the  earth's  magnetism 
Upon  a  magnetic  needle  is  merely  directive. 


LESSON  XLI.  —  Voltaic  Batteries. 

684.  What  was  the  famous  phenomenon  first  observed 
by  Galvani  in  1786,  and  how  was  it  explained  by  Galvani 
and  by  Volta  respectively  ? 

685.  Explain  the  construction  and  mode  of  action  of 
Voita's  pile. 

686.  Describe  the  arrangement  known  as  Volta? s  crown 
of  cups. 

687.  How  did  Volta  explain  the  effect  produced  by 
the  voltaic  battery  ? 

688.  Illustrate  the  manner  in  which  the  total  effect 
produced  by  Volta's  pile  depends  on  the  number  of  ele- 
ments in  the  pile. 

689.  Explain  in  what  way  the  contact  theory  as  held 
by  Volta  is  inconsistent  with  the  laws  of  energy. 

690.  What  is  the  chemical  theory  of  the  action  of  the 
voltaic  battery  ? 

691.  What  was  the  nature  of  the  crucial  experiment 
made  by  Sir  W.  Thomson  in  reference  to  the  two  theories 
of  the  voltaic  battery,  and  what  conclusions  are  to  be 
drawn  from  it  1 


62  QUESTIONS   ON  STEWARTS    [CHAP.  vin. 

692.  What  is  denoted  by  the  term  electromotive  force  ? 

693.  What  results  did  Sir  C.  Wheatstone  obtain  in  his 
experiments  on  the  electromotive  force  in  different  com- 
binations of  platinum,  zinc,  and  potassium,  and  what  gen- 
eral law  do  they  illustrate  'I 

694.  Classify  the  metals  according  to  their  order  in  the 
electromotive  series. 

695.  What  two  causes  greatly  enfeeble  a  single-liquid 
Battery  after  it  has  been  in  action  a  short  time  ? 

696.  Describe  Daniell's  constant  battery,  and  its  mode 
of  action. 

697.  What  are  the  advantages  of  amalgamating  the 
zinc  plates  ? 

6  9  a.   Describe  Grove's  battery,  and  its  mode  of  action. 

699.  How  may  the  existence  of  a  thermo-electric  cur- 
rent be  easily  demonstrated  1     Why  are  the  metals  bis- 
muth and  antimony  generally  used  in  thermo-electric 
combinations  ? 

700.  Illustrate  the  application  of  the  law  of  Art.  374 
in  thermo-electric  combinations. 

701.  Within  certain  limits  what  is  the  strength  of  a 
thermo-electric  current  proportional  to  ?     But  what  has 
Gumming  shown  in  the  case  of  copper  and  iron  ? 


LESSON  XLII.  —  Effect  of  the  Electric   Current  upon  a 
Magnet. 

702.  When,  and  by  whom,  was  the  important  dis- 
covery of  the  connection  between  an  electric  current  and 
a  magnet  made  1 

703.  Explain  the  nature  of  Oersted's  experiment. 

704.  State  the  rule  which  expresses  the  relation  be- 
tween the  behavior  of  the  needle  and  the  position  and 
direction  of  the  current,  and  apply  this  rule  to  the  four 
distinct  cases  which  are  possible. 

705.  What  is  the  object  of  a  galvanometer  ?    Explain 
the  construction  and  mode  of  action  of  a  single-needle 
galvanometer. 


LESS.  XLIIL]     ELEMENTARY  PHYSICS.  63 

706.  Explain  the  construction  and  mode  of  action  of 
an  astatic  galvanometer.    Describe  the  mirror  arrangement 
for  increasing  the  sensibility  of  a  galvanometer. 

707.  Upon  what  law  does  the  action  of  a  current  on 
a  needle  depend  ? 

708.  Describe  the  tangent  compass  and  its  action.    Why 
has  the  instrument  received  this  name  ? 

709.  Describe  an  electro-magnet.     How  do  they  com- 
pare in  strength  with  natural  magnets  1 

710.  What  curious  facts  have  been  observed  in  the 
magnetization  of  soft  iron  bars  ? 

711.  State   the    principle   of    the   electric   telegraph. 
What  takes  the  place  of  a  return  wire  in  electric  tele- 
graphs, and  what  advantages  are  gained  by  this  substitu- 
tion ?  

LESSON  XLIIL  —  Action  of  Currents  on   One  Another, 
and  Action  of  Magnets  on  Currents. 

712.  What  are  the  chief  laws  of  the  mutual  action  of 
electrical  currents  ? 

713.  Discuss  the  various  cases  which  may  arise  under 
Law  III. 

714.  Explain  a  case  in  which  a  continuous  rotation  of 
currents  is  produced  by  their  mutual  action. 

715.  Examine  the  action  of  the  earth's  magnetism  upon 
a  circular  vertical  current  which  is  free  to  place  itself  in 
any  position. 

716.  Explain  the  construction  and  behavior  of  a  sole- 
noid. 

717.  State  Ampere's  hypothesis  concerning  nlagnetism, 
and  show  that  it  explains  the  known  relations  between 
magnets  arid  currents,  and  between  magnets  and  magnets. 

718.  What  instance  of  the  conservation  of  energy  do 
we  have  in  the  case  of  two  similar  voltaic  batteries,  each 
charged  with  the  same  amount  of  zinc,  if  one  battery  is 
made  to  do  external  work,  while  the  other  does  no  exter- 
nal work  at  all  ? 


64  QUESTIONS  ON  STEWARTS  [CHAP.  vm. 

LESSON  XLIV.  —  Induction  of  Currents. 

719.  State  the  laws  of  the  induction  of  electric  cur- 
rents.    Who  discovered  current  induction  ? 

720.  Explain  how  magnets  may  be  made  to  play  the 
part  of  currents  in  the  phenomena  of  induction. 

721.  Show  that  the  phenomena  of  induction  are  in 
harmony  with  the  laws  of  energy. 

722.  Describe  a  method,  employed  by  Joule,  for  con- 
verting mechanical  energy  into  that  of  induced  currents, 
and  from  that  into  heat. 

723.  What  two  kinds  of  electrical  machines  are  there 
which  depend  for  their  action  on  the  laws  of  induction  1 

724.  What    is   the   principle    of    a    magneto-electrical 
machine  ?    What  arrangement  is  employed  in  Clark's  ma- 
chine ?    What  is  the  object  of  a  commutator  ?     For  wrhat 
purposes,  among  others,  are  these  machines  used  ? 

725.  Describe  RuhmJcorff's  coil,  and  explain  its  mode 
of  action. 


LESSON  XLV.  —  Distribution  and  Movement  of  Electricity 
in  a  Voltaic  Battery. 

726.  Who  first  developed  the  laws  regulating  the  mo- 
tion and  distribution  of  electricity  in  a  battery  ? 

727.  State  the  laws  which  regulate  the  electro-motive 
force  of  a  battery. 

728.  Investigate  the  subject  of  electrical  resistance  in 
a  manner  similar  to  that  employed  in  studying  thermal 
conductivity,  and  deduce  Ohm's  formula  for  expressing 
the  relation  between  the  intensity  of  the  current,  the  elec- 
tromotive force,  and  the  resistance  of  a  galvanic  circuit. 

729.  Upon  what  three  things  does  the  electrical  resist- 
ance of  a  substance  depend  ? 

730.  Modify  the  fundamental  formula  of  Ohm  so  as  to 
express  the  intensity  of  the  current  in  a  battery  of  ten 
cells  with  a  definite  external  resistance. 


LESS.  XLVI.]      ELEMENTARY  PHYSICS.  65 

731.  Examine,  by  means  of  Ohm's  formula,  the  effect 
of  increasing  the  number  of  cells  in  a  battery  ; 

1.  when  there  is  110  external  resistance  ; 

2.  when  the  external  resistance  is  small  compared 
with  the  internal ; 

3.  when  the   external   resistance  is  large  compared 
with  the  internal. 

732.  Why  is  it  necessary  to  have  a  large  number  of 
cells  in  order  to  produce  the  electric  light  ? 

733.  Why  is  it  advantageous  to  multiply  the  number 
of  couples  in  a  thermo-electric  current  1 

734.  Explain,  by  the  aid  of  Ohm's  formula,  the  effect 
of  increasing  the  size  of  the  plates  in  a  voltaic  battery. 

735.  What   arrangement   in    a  battery  is  preferable, 
when  the  battery  is  to  be  used  to  produce  thermal  effects  ? 
Why? 

736.  What  law,  as  to  the  intensity  of  the  current  in 
different  portions  of  a  circuit,  is  likewise  due  to  Ohm  ? 

737.  Explain  a  method  of   comparing  the  resistance 
(and  hence  the  conductivity)  of  metallic  wires  by  means 
of  a  galvanometer. 

738.  What  points  of  resemblance  have  been  observed 
between  the  electric  and   the  thermal  conductivities  of 
substances  ? 


LESSON  XLYI.  —  Effects  of  the  Electric  Current. 

739.  Compare,  as  regards  quantity,  tension  and  the 
resultant  physiological  effects,  the  Ley  den  jar  battery,  the 
voltaic  battery,  and  a  flash  of  lightning, 

740.  When  an  electric  current  is  made  to  pass  through 
a  circuit,  to  what  is  the  heating  effect  of  the  current  pro- 
portional ? 

741.  Show  that  the  increase  of  temperature  produced 
by  the  passage  of  the  same  quantity  of  electricity  through 
a  wire  will  vary  inversely  as  the  square  of  the  cross  sec- 
tion of  the  wire. 


66  QUESTIONS  ON  STEWART'S   [CHAP.  vin. 

742.  Deduce  from  the  above  law,  that  the  heat  gener- 
ated in  a  given  time  is  proportional  to  the  square  of  the 
intensity  of  the  current. 

743.  If  one  part  of  a  circuit  be  composed  of  a  metre 
of  silver  wire  two  square  millimetres  in  cross  section,  and 
another  of  five  metres  of  zine  wire  four  square  millimetres 
in  cross  section,  show  that  the  relative  heating  effects  of 
the  current  on  these  two  wires  will  be  as  1  :  8-62. 

Specific  electric  conductivity  of  silver,  100  ;  of  zinc,  29. 

744.  Looking  at  the  subject  from  the  stand-point  of 
the  doctrine  of  energy,  in  what  consists  the  difference  be- 
tween dissolving  zinc  by  acid  in  an  ordinary  vessel  and 
doing  so  by  the  voltaic  arrangement  1 

745*  Describe  how  the  electric  light  is  produced. 

746.  Define  the  terms  electrolysis,  electrolyte. 

747.  Describe   a   voltaic   arrangement   which  may  be 
employed  to  decompose  water. 

748.  What  is  the  distinction  between  electro-positive 
and  electro-negative  elements  ? 

749.  In  the  electrolytic  decomposition  of  water,  for 
example,  the  question  naturally  arises,  Is  the  oxygen  of 
each  molecule  which  is  decomposed  carried  bodily  to  the 
one  pole,  and  the  hydrogen  to  the   other  ]     What  was 
Davy's  test  experiment  upon  this  point  ? 

750.  Explain  Grotthuss's  hypothesis. 

751.  State  the  laws  of  electrolytic  action  discovered  by 
Faraday. 

752.  Explain  the  principle  of  the  electrotype  process. 

753.  What   is   the   effect   of    passing  polarized   light 
through  glass  subjected  to  the  action  of  a  powerful  electro- 
magnet ? 

754.  Under  what  conditions  are  peculiar  stratifications 
of  light  and  colors  produced  by  the  current  ? 

755.  Will  a  current  pass  through  a  perfect  vacuum  ? 

756.  What  is  the  cause  of  the  peculiar  smell  which 
is  often  noticed  when  an  electric  machine  is  in.  action  1 


LESS.  XLVII.]    ELEMENTARY  PHYSICS.  67 

CHAPTER  IX. 

ENERGY  OF  CHEMICAL    SEPARATION. 

LESSON  XLVII.  —  Concluding  Remarks. 

757.  Why  is  it  natural  to  expect  that  a  definite  amount 
of  carbon  will,  when  burnt,    always   furnish  a  definite 
amount  of  heat ? 

758.  Who  have  investigated  the  quantity  of  heat  given 
out  in  chemical  combination  ? 

759.  To  what  general  result  was  Andrew  led  by  study- 
ing the   heat  given  out  during  the  mutual  action  of 
metals  1 

760.  What  grounds  are  there  for  believing  that  the 
electro-motive  forces  are  really  those  which  cause   heat 
when  chemical  combination  takes  place  ? 

761.  What  relation  have  we  found  to  exist  between  the 
doctrine  of  the  conservation  of  energy  and  the  chimera 
of  perpetual  motion  ? 

762.  In  what  way  might  a  champion  of  perpetual  mo- 
tion assent  to  the  doctrine  of  the  conservation  of  energy 
without  absolutely  giving  up  his  cause  1 

763.  Give  the  outlines  of  the  doctrine  of  the  dissipation 
of  energy,  and  show  what  bearing  this  doctrine  has  on  the 
problem  of  perpetual  motion. 

764.  Trace  back  the  energy  of  our  system  through  its 
various  transmutations  to  its  ultimate  source. 

765.  What  vast  store  of  energy  was  provided  by  Nature 
in  geological  ages  ? 

766.  Show  that  water-power  and  wind-power  are  really 
products  of  the  sun's  rays. 

767.  What  single  small  exception  is  there  to  the  state- 
ment that  "  all  the  work  done  in  the  world  is  due  to  the 
sun"? 


68  QUESTIONS  ON  PHYSICS.        [CHAP.  ix. 

768.  What  would  seem  to  be  the  answer  which  we  must 
give  to  the  question.  Will  the  sun  last  forever  ? 

769.  In  tine,  to  what  ultimate  conclusion  does  the  prin- 
ciple of  degradation  conduct  us  1 

770.  Enumerate  the  various  kinds  of   energy  which 
have  been  studied  in  this  book. 

771.  Recapitulate  various  instances  of  the  transmuta- 
tion of  visible  kinetic  energy. 

772.  Give  examples  of  the  conversion  of  visible  poten- 
tial energy. 

773.  Enumerate  the  instances  of  the  transmutation  of 
heat. 

774.  Give  instances  of  the  transmutation  of  radiant 
energy. 

775.  Mention  examples  of  the  transformation  of  the 
energy  of  electrical  separation. 

776.  Give  examples  of  the  conversion  of  the  energy  of 
electricity  in  motion. 

777.  Give  instances  of  the  transmutation  of  the  energy 
of  chemical  separation. 


PART  II. 
EXERCISES  AND  PROBLEMS. 


THE  Exercises  in  large  type  are,  in  the  main,  direct  and  simple  appli- 
cations of,  or  deductions  from,  the  principles  of  the  text-book :  in  the 
cases  in  which  special  difficulties  might  arise  or  in  which  new  definitions 
are  introduced,  hints  or  explanations  will  be  found  in  Part  III.  (Answers 
and  Solutions).  These  Exercises  demand  only  a  fair  knowledge  of  the 
Elements  of  Arithmetic,  Algebra,  and  Plane  Geometry. 

The  Exercises  printed  in  smaller  type  are  intended  to  be  more  difficult 
than  the  others,  and  some  of  them  involve  principles  which  are  not  ex- 
plicitly stated  in  the  text-book.  They  are  designed  chiefly  for  use  with 
advanced  sections  or  as  voluntary  exercises.  Those  who  wish  to  take 
them  should  have  an  elementary  knowledge  of  Plane  Trigonometry  and 
of  Analytic  Geometry.  In  a  few  cases  a  knowledge  of  the  Calculus  may 
perhaps  be  serviceable,  although  it  is  not  required.  A  summary  of 
mathematical  data  and  formulae  is  given  in  Appendix  IV.  When  aid  is 
required,  it  must  be  obtained  from  competent  teachers,  or  from  books. 
Appended  is  a  list  of  elementary  works  which  may  be  consulted  with 
advantage,  particularly  the  first  two  and  the  last  two. 

THOMSON  AND  TAIT'S  Elements  of  Natural  Philosophy,  Part  I.  (Lon- 
don &  New  York  :  McMillan  &  Co.) 

KERR'S  Rational  Mechanics.     (Glasgow:  W.  Hamilton.) 

GOODWIN'S  Elementary  Statics,  and  Elementary  Dynamics.  (Cambridge, 
England  :  Deighton,  Bell,  &  Co.) 

BESANT'S  Elementary  Hydrostatics.  (Cambridge,  England :  Deighton, 
Bell,  &  Co.) 

HAUGHTON'S  Manual  of  Mechanics.  (London  &  New  York :  Cassell, 
Fetter,  &  Galpin.) 

TODHUNTEB'S  Mechanics  For  Beginners.  (London  &  New  York  :  McMil- 
lan &  Co.) 

GOODBYE'S  Principles  of  Mechanics.  (London:  Longmans,  Green, 
&Co.) 

BURAT,  Precis  de  Mecanique.     (Paris  :  Victor  Masson  et  Fils.) 

BRIOT,  Lecons  de  Mecanique.    (Paris  :  Dunod,  Editeur.) 

BRESSE  ET  ANDRE,  Cours  de  Physique,  les  2  premier  fascicules.  (Paris  : 
Dunod,  Editeur.) 

DESCHANNEL'S  Natural  Philosophy,  Translated  by  EVERETT,  Part  I. 
(New  York  ;  D.  Appleton  &  Co.) 


70  ELEMENTARY  PHYSICS.  [CHAP.  I. 


INTRODUCTION. 


1.  Give  an  illustration,  not  mentioned  by  the  author, 
of  relative  motion. 

2.  Give  an  illustration,  not  mentioned  by  the  author, 
of  force  producing  motion  ;  also,  of  force  stopping  motion. 

3.  Give  an  example  of  forces  in  equilibrium. 

4.  If  you  are  running  towards  the  North,  and,  as  sud- 
denly' as  possible,  change  the  direction  of  your  motion 
from  the  North  to  the  East,  do  you  think  that  force  is 
expended  in  producing  this  change  ? 

5.  Can  you  give  an  instance  of  a  body  which  is  not 
acted  upon  by  any  force  whatever  1 

6.  Mention  an  object  which  is  known  to  us  through  the  medium 
of  a  single  sense  ;  also,  an  object  which  is  known  through  the 
medium  of  more  than  one  sense. 

7.  Explain  and  illustrate  the  distinction  between  a  phenomenon 
and  a  law  of  Nature. 

8.  What  distinction  can  you  draw  between  a  body  and  a  sub- 
stance. 

9.  Two  steamers  are  moving  with  equal  velocities  in  the  same 
direction.     A  passenger  on  one  steamer  looks  at  the  other  from  his 
state-room  window  ;  how  will  it  appear  to  him  ?    Suppose  the 
other  steamer  suddenly  appears  to  change  its  velocity ;  in  what 
two  ways  might  this  phenomenon  be  produced  ? 


LESS,  i.]         EXERCISES  AND  PROBLEMS.  71 


CHAPTER  I. 


LAWS    OF   MOTION. 

LESSON  I.  —  Determination  of  Units. 

10.  How  many  square  feet  are  there  in  124  acres  ? 

11.  How  many  square  decimetres  are  there  in  124 
ares  ? 

12.  Reduce  346768595  cubic  inches  to  cubic  yards. 

13.  Reduce   346768595    cubic    centimetres    to    cubic 
metres. 

14.  How  many  litres  are  there  in  2  steres  1 

15.  Reduce  6,000,000  grammes  to  tonnes. 

16.  Reduce  218*75  grains  to  grammes. 

17.  What  ratio  exists  between  a  cubic  centimetre  and  a 
cubic  metre  1 

18.  What  is  the  weight  of  64  litres  of  water  1  of  64 
cubic  centimetres  of  water  ? 

19.  Show  that  the  number  which  expresses  the  volume 
in  litres  of  a  quantity  of  water  also  denotes  the  mass  in 
kilogrammes.     Examine  also  the  case  in  which  the  volume 
of  the  water  is  expressed  in  cubic  centimetres. 

20.  A  rectangular  trough  is  12  metres  long,  2  metres 
wide,  and  80  centimetres  deep.     How  many  kilogrammes 
of  water  will  it  hold  1 

21.  Define   density.     What  is  the  numerical  measure 
of  the  density  of  a  substance  ? 

22.  Prove  tkat  the  mass  of  a  body  is  equal  to  the  pro- 
duct of  its  volume  and  its  density  ;  or,  if  V  denotes  the 
volume,  D  the  density,  and  M  the  mass,  that  M  —  VD. 

23.  Why  is  the  density  of  water  equal  to  unity  in  the 
Metric  System  ? 

24.  Find  the  mass  of  74  litres  of  cork  (density  of  cork, 
0.24). 


72  ELEMENTARY  PHYSICS.  [CHAP.  i. 

25.  Prove  that  the  densities  of  two  bodies  are  propor- 
tional to  the  masses  of  equal  volumes  of  the  bodies. 

26.  Explain  the  distinction  between  mass  and  weight. 

27.  A  ship  sails  504  miles  in  a  week.    Find  the  average 
velocity  in  miles  per  hour. 

28.  Compare  the  velocities  of  two  points  which  move 
uniformly,  one  through  5  feet  in  half  a  second,  the  other 
through  100  yards  in  a  minute. 

29.  The  daily  rotation  of  the  earth  is  uniform.    Taking 
its  circumference  as  25,000  miles,  determine  the  velocity 
of  a  point  on  the  equator. 


30.  A  body  is  moving  with  a  velocity  of  30  feet  per  second. 
With  what  velocity  must  another  body  move,  which  starts  from 
a  given  point  3  minutes  after  the  former  and  overtakes  it  in  10 
minutes  ? 

31.  For  6  seconds  a  body  moves  with  a  velocity  of  10,  and  for 
the  next  9  seconds  with  a  velocity  of  15.     What  uniform  velocity 
would  have  carried  it  over  the  same  space  in  the  same  time  ? 

32.  Compare  the  velocities  of  two  points,  one  of  which  moves 
uniformly  around  the  circumference  of  a  circle  in  the  same  time 
that  the  other  moves  along  the  diameter. 

33.  The  height  of  a  cylindrical  cistern  is  12  metres  and  its  di- 
ameter is  6*5  metres.     How  many  kilogrammes  of  water  will  it 
hold  ? 

34.  One  litre  of  a  substance  weighs  280  grammes,  and  a  piece 
of  another  substance  twice,  as   dense   as  the    first    weighs    400 
grammes.     Find  the  volume  of  the  second  substance. 

35.  Find  the  ratio  of  the  kilometre  to  the  nautical  mile  or  knot. 

36.  Show  that  the  proper  measure  of  density  is  the  mass  of 
unit  of  volume. 

37.  If  the  unit  of  mass  be  increased  a  times,  and  the  unit  of 
volume  be  increased  b  times,  how  will  the  measure  of  density  he 
altered  ? 

38.  A  cylindrical  log  of  wood,  a  metres  long,  and  b  centimetres 
in  diameter,  weighs  c  kilogrammes.     Compare  its  density  with  that 
of  a  substance  the  density  of  which  is  known  to  be  d. 

39.  What  are  the  dimensions  of  velocity  in  terms  of  the  funda- 
mental units  of  length  and  time. 

40.  Define  angular  velocity.     What  is  the  unit  of  angular  ve- 
locity, and  what  are  its  dimensions  in  terms  of  the  fundamental 
units  ? 


LESS,  ii.]        EXERCISES  AND  PROBLEMS.  73 

41.  Find  the  linear  velocity  with  which  a  point  must  move  on 
the  circumference  of  a  circle  in  order  to  describe  one  unit  of  angu- 
lar velocity  per  second. 

42.  What  is  the  measure  of  the  angular  velocity  of  the  hour 
hand  of  a  clock  ?  of  the  minute  hand  '( 

4-3.  Prove  that  the  linear  and  angular  velocities  of  a  point  mov- 
ing on  the  circumference  of  a  circle  are  connected  by  the  equation, 
v  —  r  co,  in  which  v  denotes  linear  velocity,  r  radius  of  circle,  and 
co  angular  velocity. 

44.  Two  bodies  begin  to  move  uniformly  at  the  same  time  along 
the  same  line,  the  first  from  a  point  A  with  a  velocity  v,  the  second 
from  a  point  B  with  a  velocity  vf  :  — 

(1)  How  far  apart  will  they  be  at  the  end  of  t  seconds  ? 

(2)  When  will  they  be  together  ? 

(3)  How  far  will  they  be  from  A  when  they  are  together  ? 

45.  If  a  velocity  be  expressed  by  6  when  one  second  is  taken  as 
the  unit  of  time,  what  would  be  its  measure  if  one  minute  were 
taken  as  the  unit  of  time  ? 

46.  If  v  denotes  a  velocity  in  the  metre-second  system,  prove 

that  the  same  velocity  will  be  denoted  by  — ,  in  a  system  in  which 

the  unit  of  length  is  m  metres  and  the  unit  of  time  n  seconds. 

47.  Prove  that  if  two  points  move  uniformly  with  any  velocities 
in  fixed  directions,  the  line  joining  the  points  will  always  remain 
parallel  to  itself. 


LESSON  II.  —  First  Law  of  Motion. 

48.  When  we  find  a  body  moving  uniformly  and  in 
one  constant  direction,  what  may  we  infer  with  regard  to 
the  total  force  that  is  acting  upon  the  body  ? 

49.  Illustrate  the  principle  of  Inertia  by  reference  to 
the  condition  in  which  a  person  finds  himself  when  stand- 
ing in  a  boat  at  starting  or  stopping. 

50.  Account  for  the  practical  rule  which  habit  teaches 
us  to  observe  in  jumping  from  a  carriage  which  is  in  mo- 
tion. 

51.  Serious  accidents  have  sometimes  happened  by  car- 
riages oversetting  when  moving  along  a  sharp  curve  in 


74  ELEMENTARY  PHYSICS.          [CHAP.  I. 

the  road.     Explain  the  cause  of  these  accidents,  and  show 
how  they  might  have  been  prevented. 

52.  Show  that  the  First  Law  of  Motion  contains  the  convention 
•universally  adopted  for  the  measurement  of  Time. 

53.  What  definition  of  Force  does  the  First  Law  of  Motion  give 
us? 

54.  Review  briefly  the  evidence  in  favor  of  the  truth  of  the  First 
Law  of  Motion. 


LESSON  III.  —  Second  Law  of  Motion.    Motion  produced 
by  Gravity.     Kinematics. 

[In  the  problems  upon  the  motion  produced  by  gravity  the  resistance 
of  the  air  is  neglected,  and  g  is  to  be  taken  as, equal  to  32 '2  feet,  or  9 '8 
metres.] 

55.  Give  an  additional  illustration  of  the  action  of  a 
single  force  on  a  moving  body. 

56.  State  in  general  terms  the  rule  for  compounding 
two  simultaneous  motions  or  velocities  in  different  direc- 
tions, —  a  rule  or  proposition  known  as  the  Parallelogram 
of  Velocities,  —  and  give  a  demonstration  of  the  same. 

57.  If  a  man  is  rowing  a  boat  directly  across  a  river 
two  miles  wide  at  the  rate  of  four  miles  an  hour,  and  the 
current  at  the  same  time  is  taking  the  boat  down  stream 
at  the  rate  of  three  miles  an  hour,  find,  — 

(1)  In  what  direction  the  boat  will  move  ; 

(2)  How  far  it  will  have  gone  when  it  reaches  the 
opposite  bank  ; 

(3)  How  far  the  landing-place  will  be  from  the  point 
directly  opposite  the  starting-place  ; 

How  long  the  boat  is  in  motion  ; 

How  long  it  would  have  taken  to  cross  the  river 

if  there  had  been  no  current. 

58.  Explain  the  geometrical  method  of  finding  a  single 
velocity  equivalent  to  any  number  of  simultaneous  veloci- 
ties. 


LESS,  in.]      EXERCISES  AND  PROBLEMS.  75 

59.  Explain  the  geometrical  method  of  resolving  a  ve- 
locity in  a  given  direction  into  two  component  velocities  in 
any  given  directions. 

60.  Prove  that  if  velocities  represented  by  the  sides  of 
a  triangle  taken  in  the  same  order  be  impressed  simultane- 
ously upon  a  point  it  will  remain  at  rest. 

61.  Prove  that  the  resultant  of  velocities  represented 
by  the  sides  of  any  closed  polygon  whatever,  taken  all  in 
the  same_prder,  is  zero. 

62.  Show  that  the  value,,  or  resolved  part,  or  effective 
component  of  a  known  velocity,  estimated  along  a  given 
line,  is  the  projection  of  the  line  representing  the  velocity 
upon  the  given  line.     Examine  the  case  in  which  the 
given  line  makes  a  right  angle  with  the  line  representing 
the  velocity. 

63.  Suppose  that  six  forces  act  simultaneously  upon  a 
body,  such  that  separately  they  would  impart  to  it  the  fol- 
lowing velocities  :  — 

4  feet  per  second  towards  the  East, 


North, 
tt        a 

«    West, 

u        tt 

"    South. 


Find  the  magnitude  and  direction  of  the  resultant  ve- 
locity. 

64.  If  a  cannon-ball  were  discharged  from  the  rear  end 
of  an  express-train,  directly  along  the  track,  at  the  same 
rate  as  the  train  is  moving  forwards,  what  would  be  the 
motion  of  the  ball  relative  to  the  ground  'I 

65.  Expose  the  fallacy  in  the  following  specimen  of 
erroneous  mechanical  reasoning  :  — 

"Let  the  ball  be  thrown  upwards  from  the  mast-head  of  a 
stationary  ship,  and  it  will  fall  back  to  the  mast-head,  and  pass 
downwards  to  the  foot  of  the  mast.  The  same  result  would 
follow  if  the  ball  were  thrown  upwards  from  the  mouth  of  a 
mine,  or  the  top  of  a  tower,  on  a  stationary  earth.  Now  put  the 


76 


ELEMENTARY  PHYSICS. 


[CHAP.  i. 


ship  in  motion,  and  let  the  ball  be  thrown  upwards.  It  will,  as 
in  the  first  instance,  partake  of  the  two  motions,  —  the  upward  or 
vertical  A  C,  and  the  horizontal  A  B,  as  shown  in  Fig.  47 ;  but 


because  the  two  motions  act  conjointly,  the  ball  will  take  the 
diagonal  direction  A  D.  By  the  time  the  ball  has  arrived  at 
D,  the  ship  will  have  reached  the  position  B;  and  now,  as  the 
two  forces  will  have  been  expended,  the  ball  will  begin  to  fall, 
by  the  force  of  gravity  alone,  in  the  vertical  direction  D  B  II  ; 
but  during  its  fall  towards  H,  the  ship  will  have  passed  on  to  the 
position  S,  leaving  the  ball  at  H,  a  given  distance  behind  it."  * 

66.  Bishop  Wilkins,  an  English  divine  of  the  17th 
century,  and  author  of  a  Treatise  on  the  Art  of  Flying, 
proposed  the  following  "  new  and  easy  way  of  travelling." 
A  large  balloon  was  to  be  constructed  and  provided  with 
apparatus  to  work  against  the  varying  currents  of  the  air. 
The  balloon,  having  been  allowed  to  ascend  to  a  conven- 
ient height,  was  to  be  kept  practically  at  rest  by  working 
the  apparatus,  while  the  earth  revolved  beneath  it  ;  and. 
when  the  desired  locality  came  in  view,  those  in  the  bal- 

*  Earth  Not  a  Globe,  by  "  Parallax,"  pp.  64,  65.  London  : 
John  B.  Day.  1873. 


LESS,  in.]      EXERCISES  AND  PROBLEMS.  77 

loon  were  to  let  out  gas  and  drop  down  at  once  to  the 
earth's  surface.  In  this  way  New  York,  for  example, 
would  be  reached  from  London  in  a  few  hours,  or  rather 
New  York  would  reach  the  balloon  at  the  rate  of  more 
than  700  miles  an  hour. 

Show  the  futility  of  any  such  method  of  travelling. 

67.  Define  acceleration,  uniform  acceleration,  variable 
acceleration.     How  is  uniform  acceleration  measured  ? 

68.  Find  the   average  acceleration  of  a  point  the  ve- 
locity of  which  increases  from  ten  miles  per  hour  to  sixty 
miles  per  hour  in  two  hours. 

If  at  the  end  of  the  first  hour  the  velocity  is  fifty  miles 
per  hour,  find  the  average  acceleration  during  each  hour. 

69.  A  constant  force  acting  thirteen  seconds  produces 
a  velocity  of  four  miles  per  hour.      Find  the  accelera- 
tion. 

70.  A  body  falls  to  the  ground  from  rest  in  6  seconds  ; 
find  the  space  passed  over. 

71.  Through  what  space  will  a  body  fall  in  the  ninth 
second  of  its  descent  ? 

72.  A  stone  strikes  the  ground  with  a  velocity  of  98 
metres  per  second.     Find  the  height  fallen  through. 

73.  How  long  must  %  a  body  fall  to  acquire  a  velocity 
of  322,  feet  per  second  ? 

r  74.  -A  rocket  begins  to  ascend  vertically  with  a  velocity 
of  161  feet  per  second.  How  high  will  it  rise,  and  what 
time  will  be  ocoupied  in  both  ascent  and  descent  1 
•"  75.  Deduce  and  explain  the  general  formula  for  deter- 
mining the  final  velocity  v  of  a  body  which,  having  an 
initial  velocity  a,  is  acted  on  by  gravity  for  t  seconds, 

viz.  :  — 

v  —  a  +  g  t. 

76.  A  body  is  thrown  downward  with  a  velocity  of 
160  feet  per  second  ;  find  its  velocity  at  the  end  of  five 
seconds. 

77.  A  body  is  thrown  upward  with  a  velocity  of  49 
metres  per  second  ;  with  what  velocity  and  in  what  di- 
rection will  it  be  moving  at  the  end  of  7  seconds  1 


78  ELEMENTARY  PHYSICS.  [CHAP.  r. 

78.  If  a  body  have  an  initial  velocity  «,  prove  that  the 
space  passed  over  in  t  seconds  under  the  action  of  gravity 
is  given  by  the  equation,  — 

s  =  at  +  \gt\ 

79.  From  the  formulae  of  Exercises  75  and  78  deduce 
the  following,  — 

v'2  =  2g-s  +  a2 
s  =  ^  (v  +  a)  t 

80.  Prove  that  the  time  of  ascent  of  a  body  projected 
vertically  upward  with  the  velocity  a  is  ?,  and  that  the 

9 

height  ascended  is 

81.  Prove  that  the  times  of  ascent  and  of  descent  of  a 
body  thrown  upward  are  the  same. 

82.  Prove  that  if  a  body  is  projected  vertically  upward 
it  returns  to  the  ground  with  the  same  velocity  as  that 
with  which  it  was  projected. 

83.  A  cannon-ball,  fired  vertically  upward,  returned 
to  the  ground  in  20  seconds  ;  find  the  height  ascended 
and  the  velocity  of  projection. 

84.  With  what  velocity  must  a  stone  be  thrown  down 
a  well  100  metres  deep,  in  order  that  it  may  reach  the 
bottom  of  the  well  in  one  second  ? 

85.  Analyze  geometrically  the  action  of  the  wind  on  the  sails  of 
a  vessel,  and  explain  how  it  is  possible  for  a  vessel  to  sail  nearly 
against  the  wind. 

86.  Explain  the  action  of  the  wind  on  a  kite. 

87.  A  man  can  row  a  boat  at  a  certain  rate,  and  the  current  of 
a  river  is  flowing  at  a  certain  rate  ;  find  the  direction  in  which  the 
boat  should  be  steered  in  order  that  it  may  be  rowed  directly  across 
the  river. 

88.  Solve  the  problem  of  compounding  two  velocities  by  the 
trigonometrical  method.     Let  u  and  v  denote  the  velocities,  w 
their  resultant,  $,  a,  ft,  the  angles  between  the  direction  of  u  and  v, 
u  and  w,  v  and  w,  respectively  ;  then  the  problem  is,  —  given  u,  v, 
and  <J>,  prove  that,  — 


LESS,  in.]       EXERCISES  AND  PROBLEMS.  79 

w2  =  u*  +  vz  +  2  u  v  cos  <£. 

=  —  sin  <£. 
w 

sin  8  =  !i  sin  0. 
«0 

89.  Deduce  formulae  from  those  in  the  preceding  problem  for 
resolving  a  given  velocity  (w)  into  two  velocities  (w,  IT),  making  any 
assigned  angles  (a,  /3)  with  the  given  velocity. 

90.  The  value  of  two  velocities  are  36  and  60,  and  the  angle 
between  their  directions  is  54° ;  find  the  resultant  velocity,  and 
the  angles  which  its  direction  makes  with  those  of  the  given  ve- 
locities. 

91.  Two  equal  velocities  are  simultaneously  impressed  upon  a 
body,  one  towards  the  north,  the  other  towards  the  east.     The 
resultant  velocity  is  equal  to  10.     Find  the  two-velocities,  and  the 
direction  in  which  the  body  will  move. 

92.  Explain,  on  kiuematical  principles,  why  the  northern  trade- 
wind  appears  to  blow  from  a  northeasterly  direction,  and  the 
southern  trade-wind  from  a  southeasterly  direction. 

93.  A  person  travelling  due  east  at  the  rate  of  4  miles  an  hour 
observes  that  the  wind  seems  to  blow  directly  from  the  south  ;  and 
that,  on  doubling  his  speed,  it  appears  to  blow  from  the  southeast. 
Find  the  velocity  and  the  direction  of  the  wind. 

94.  A  deer  is  running  at  the  rate  of  20  miles  an  hour,  and  a 
sportsman  fires  at  him  when  he  is  at  the  nearest  point,  200  yards 
distant ;  what  allowance  should  be  made  in  taking  aim,  'supposing 
the  velocity  of  the  rifle  bullet  to  be  1000  feet  per  second  ? 

95.  A  particle  descends  vertically  T&long  the  axis  of  a  tube 
which  at  the  same  time  is  carried  forward  in  the  horizontal  direc- 
tion, both  motions  being  uniform  ;  find  the  inclination  of  the  tube 
from  the  vertical  line. 

96.  Show  that,  in  the  case  of  variable  velocity,  the  equation 

v  =  £  (which  expresses  the  definition  of  the  average  velocity  for 

any  time)  is  more  and  more  nearly  true  as  the  interval  of  time  is 
taken  smaller  and  smaller.  Thence  obtain  the  true  measure  of 
variable  velocity. 

97.  Prove  that  the  dimensions  of  acceleration  are  — - . 

98.  If  g  denote'  an  acceleration  when  the  second  is  the  unit  of 
time  and  the  foot  is  the  unit  of  length,  then,  if  we  take  m  seconds 
as  the  unit  of  time  and  n  feet  as  the  unit  of  length,  the  same  ac~ 

celeration  will  be  denoted  by  2!L  g. 
n 


80  ELEMENTARY  PHYSICS.  [CHAP.  i. 

99.  Find  the  measure  of  the  acceleration  of  gravity  when  one 
minute  is  taken  as  the  unit  of  time. 

100.  A  stone  is  thrown  vertically  upward  with  a  velocity  3  g  ; 
find  at  what  time  its  height  will  be  4  g,  and  its  velocity  at  this 
time. 

101.  A  stone  is  dropped  into  a  well,  and  is  heard  to  strike  the 
surface  of  the  water  after  4*5  seconds  ;  find  the  distance  to  the 
surface  of  the  water,  knowing  that  the  velocity  of  sound  is  340 
metres  per  second. 

102.  A  body  is  dropped  from  a  height  of  100  feet,  and  at  the 
same  moment  another  body  is  projected  vertically  from  the  ground: 
they  meet  half-way.     What  was  the  velocity  of  projection  of  the 
second  body  ? 

103.  A   body  is  projected  vertically  with  a  velocity  of    30 
metres  per  second.      A  second  later  another  body  is  projected 
vertically  from  the  same  point  with  a  velocity  of  40  metres  per 
second.     When  and  where  will  the  two  meet  ? 

104.  With  what  velocity  must  a  body  be  projected  downwards 
that  in  n  seconds  it  may  overtake  another  body  which  has  already 
fallen  from  the  same  point  through  a  distance  of  a  feet  ? 

105.  Prove  that,  when  we  take  into  account  the  resistance  of  the 
air,  the  time  of  ascent  of  a  body  projected  vertically  upward  is  less 
than  the  time  of  descent ;  and  that  the  velocity  on  reaching  the 
ground  is  less  than  the  velocity  on  starting. 

106.  Prove  that  the  velocity  acquired  in  sliding  down  a  smooth 
inclined  plane  is  the  same  that  would  be  acquired  in  falling  freely 
through  the  vertical  height  of  the  plane. 

107.  Prove  that  the  time  of  falling  from  rest  down  any  chord 
of  a  vertical  circle,  drawn  either  from  the  highest  or  the  lowest 
point  of  the  circle,  is  constant. 

108.  Find  the  straight  line  of  quickest  descent  from  a  given 
point  to  a  given  straight  line. 

109.  If  a  be  the  base  of  an  inclined  plane,  find  the  height  in 
order  that  the  time  of  descent  may  be  a  minimum. 

PROJECTILES. 

110.  Prove  that  a  body  projected  in  any  direction  not  vertical, 
and  acted  on  by  gravity,  will  describe  a  parabola. 

111.  Find  the  velocity  at  any  point  of  the  path  of  a  projec- 
tile. 

112.  Determine  the  position  of  the  focus  of  the  parabola  de- 
scribed by  a  projectile. 

113.  Determine  the  latus  rectum  of  the  parabola  described  by  a 
projectile. 


LKSS.  IIL]      EXERCISES  AND  PROBLEMS.  81 

114.  Find  the  maximum  height  reached  by  a  projectile. 

115.  Find  the  whole  time  of  flight  of  a  projectile,  and  show  that 
the  times  of  ascent  and  of  descent  are  equal. 

116.  Determine  the  range  of  a  projectile  on  a  horizontal  plane 
-  through  the  point  of  projection. 

117.  Prove  that  the  maximum  range  along  a  horizontal  plane 
for  a  given  velocity  of  projection  is  obtained  by  making  the  angle 
of  projection  equal  to  45°,  and  is  equal  to  the  square  of  the  velocity 
divided  by  g. 

118.  Find  the  horizontal  range  of  a  shell  fired  at  an  angle  of 
45°  with  a  velocity  of  500  feet  per  second. 

119.  Two  bodies  are  projected  simultaneously  from  the  same 
point  with   different  velocities  and  in  different  directions  ;  find 
their  distance  apart  at  the  end  of  a  given  time. 

120.  Find  the  velocity  and  direction  of  projection  in  order  that 
a  projectile  may  pass  horizontally  through  a  given  point. 

121.  Find  the  velocity  with  which  a  body  must  be  projected  in 
a  given  direction  from  the  top  of  a  tower  so  as  to  strike  the  ground 
at  a  given  point. 

122.  A  shell  is  to  be  fired  from  the  top  of  a  cliff  300  feet  high 
with  a  velocity  of  600  feet  per  second,   so  as  to  strike  a  ship  at 
anchor  600  yards  from  the  base  of  the  cliff.     What  must  be  the 
elevation  of  the  gun  ? 

123.  Two  bodies  are  projected  from  the  top  of  a  tower  at  the 
same  instant,  the  one  vertically  upward  with  a  velocity  of  100  feet 
per  second,  and  the  other  horizontally  with  a  velocity  of  60  feet 
per  second ;  find  their  distance  apart  at  the  end  of  2  seconds. 

CURVILINEAR   MOTION. 

124.  Taking  acceleration  in  its  expanded  sense,  viz.,  rate  of 
change  of  velocity  whether  the  change  takes  place  in  tlie  direction 
of  motion  or  not,  illustrate  what  is  meant  by  change  of  velocity, 
and  show  how  a  curve  may  be  drawn  which  shall  represent  the 
direction  of  the  acceleration  of  a  moving  point  at  every  instant. 

125.  If  a  point  describe  a  circle,  of  radius  r,  with  the  uniform 
velocity  v,  prove  that  the  acceleration  is  directed  towards  the 

centre,  and  is  equal  to  -  . 
r 

126.  A  stone  is  whirled  round  at  the  end  of  a  string  2  metres 
long  with  a  velocity  of  12  metres  per  second  ;  find  the  acceleration. 

127.  If  T  denote  the  time  of  revolution  or  period  of  a  point 
which  describes  a  circle  with  uniform  velocity  v,  prove  that  — 

acceleration  4  *"2 

radius         ~    T^ ' 
4*  P 


82  ELEMENTARY  PHYSICS.  [CHAP.  I. 

128.  If  <o  denote  angular  velocity  in  uniform  circular  motion, 
and  T  the  period,  prove  that 

2  _  4  7T2        acceleration 
~W  =         radius 

129.  Distinguish  between  tangential  and  normal  acceleration. 
Examine  and  illustrate  the  cases,  (1)  in  which  there  is  tangential 
but  no  normal  acceleration,  (2)  in  which  there  is  normal  but  no 
tangential  acceleration,  (3)  in  which  there  is  neither  tangential  nor 
normal  acceleration. 

SIMPLE   HARMONIC   MOTION. 

130.  Define  simple  harmonic  motion.     Give  illustrations. 

131.  Define  the  following  terms  relating  to  simple  harmonic 
motion,  —  amplitude,  displacement,  period,  phase,  epoch. 

132.  Prove  that  the  velocity  of   a  point   executing  a   simple 
harmonic  motion  is  at  any  instant  directly  proportional  to  the  dis- 
placement at  a  quarter  of  a  period  earlier  in  phase. 

When  is  the  velocity  a  maximum  ?     When  a  minimum  ? 

133.  Prove  that  the  acceleration  of  a  point  executing  a  simple 
harmonic  motion  is  at  any  instant  directed  towards  the  middle 
point,  and  is  directly  proportional  to  the  displacement. 

When  is  the  acceleration  a  maximum  ?     When  a  minimum  ? 

134.  If  T  denote  the  period  of  a  simple  harmonic  motion,  and 
to  the  angular  velocity  on  the  circle  of  reference,  prove  that  at  any 
time,  — 

acceleration         4  n-2 
displacement  ~    T2 


LESSON  IV.  —  Second  Law  of  Motion.     Measure  of  Force. 

135.  State  the  Second  Law  of  Motion  in  the  exact  lan- 
guage employed  by  Newton. 

136.  Explain  the  meaning  of  the  words  "change  of 
motion  "  in  Newton's  Second  Law. 

137.  State,  in  the  most  general  form,  the  Principle  of 
the  Independence  of  the  Effects  of  Forces,  which  is  con- 
tained in  the  second  clause  of  the  Second  Law  of  Motion 
as  given  by  Newton.    Give  examples  which  tend  to  estab- 
lish the  truth  of  this  Principle. 


LESS,  iv.]      EXERCISES  AND  PROBLEMS.  83 

138.  Why  does  it  follow  from  the  Second  Law  that 
the  proper  dynamical  measure  of  a  force  is  the  product  of 
mass  and  acceleration,  or  the  momentum  generated  in  unit 
of  time. 

139.  Define  the  dynamical  unit  of  force,  first  in  gen- 
eral terms,  and  then  in  terms  which  express  the  British 
and  the  Metric  values  respectively. 

140.  Find  the  ratio  between  the  British  and  Metric 
dynamical  units  of  force. 

141.  Prove  that  one  dynamical  unit  of  force  is  equal  to 
one  statical  unit  of  force  (the  pound  or  kilogramme)  di- 
vided by  the  value  of  g  at  the  given  locality.     Hence 
derive  rules  for  reducing  statical  forces  or  pressures  to 
dynamical  measure,  and  vice  versa. 

142.  Distinguish  between  the  two  uses   of  the  word 
pound  (or  kilogramme). 

143.  Explain  the  employment  of  spring  balances  and 
common  balances  in  estimating  masses  and  forces. 

144.  The   dynamical   measure   of  a  force  (in   Metric 
units)  is  7252  ;  reduce  this  to  Paris  kilogrammes  of  force 
(at  Paris,  g  =  9'81  metres). 

145.  A  mass  of  80  kilogrammes  at  Paris  is  moved  by  a 
constant  force  which  generates  in  one  second  a  velocity  of 
6  metres  per  second.     Find  the  measure  of  the  force,  (1)  in, 
dynamical  units,  (2)  in  kilogrammes. 

146.  On   a  mass  of   140   Ibs.  at   New  York   (where 
g  =  32*16  feet)  a  force  of  10  Ibs.  is  constantly  acting ; 
find  the  acceleration. 

147.  The  mass  of  an  iron  ball  in  pounds  is  m.     Find 
its  mass  in  kilogrammes  ;  its  weight  at  London  in  British 
statical  units  ;  its  weight  at  Paris  in  Metric  statical  units  ; 
its  weight  in  British  dynamical  units  at  London,  where 
g  —  32-19  feet  ;  its  weight  in  Metric  dynamical  units  at 
Paris,  where  g  —  9'81  metres. 

148.  From  what  height  must  the  ram  of  a  pile-driver, 
weighing  16  cwt.,  descend  upon  the  head  of  a  pile,  in  order 
that  it  may  strike  it  with  a  momentum  equal  to  that  of  a 
42  Ib.  shot,  fired  with  a  velocity  of  1610  feet  per  second  1 


84  ELEMENTARY  PHYSICS.  [CHAP.  i. 

149.  Given  the  mass  and  acceleration  of  a  body,  find 
the  rate  at  which  the  momentum  of  the  body  is  changing. 

150.  A  pressure  of  2  tons  acts  upon  a  railway  train  for 
10  minutes  ;  find  the  momentum  acquired  by  the  train. 

151.  How  long  must  a  force  of  100  kilogrammes  act  on 
a  mass  of  2000  kilogrammes  to  impress  upon  it  a  velocity 
of  5  metres  per  second  ?     What  must  be  the  magnitude 
of  the  force  which  would  bring  the  body  to  rest  in  one 
second  1 

152.  Prove  that  two  forces  are  to  each  other  as  the  ac- 
celerations which  they  impress  upon  the  same  or  upon 
equal  masses. 

153.  Show  that  the  following  proposition  follows  im- 
mediately from  the  Second  Law  :    The  resultant  of  any 
number  of  forces  applied  at  one  point  is  found  by  the  same 
geometrical  process  as  the  resultant  of  any  number  of  simul- 
taneous velocities.     What  is  this  geometrical  process  ? 

154.  Review  the  evidence  in  favor  of  the  Second  Law  of  Motion. 

155.  Review  the  course  of  reasoning  by  which  the  Second  Law 
conducts  us  to  the  measure  of  force. 

156.  Explain  how  the  Second  Law  also  gives  us  the  means  of 
measuring  mass. 

157.  In  what  three  ways  may  the  quantity  g  he  defined  ? 

158.  Show  that  the  dimensions  of  momentum  are . 

159.  Show  that  the  dimensions  of  the  dynamical  unit  of  force  are 
ML 

T%  ' 

160.  A  body  weighing  m  Ibs.   is  moved  by  a  constant  force 
which  generates  in  the  body  in  one  second  a  velocity  of  a  feet  per 
second;  find  the  weight  which  the  force  could  support. 

161.  Find  in  what  time  a  force  which  would  support  against 
gravity  a  mass  of  4  Ibs.  would  move  a  mass  of  9  Ibs.  through 
49  feet  along  a  smooth  horizontal  plane.     Find  also  the  velocity 
acquired. 

162.  Find  how  far  a  force  which  would  support  a  mass  of  n 
units  would  move  a  mass  of  m  units  in  t  seconds  ;  and  find  the 
velocity  acquired. 

163.  A  weight  (mass)  of  8   Ibs.    hanging  vertically  draws   a 
weight  (mass)  of  12  Ibs.  by  means  of  a  string,   passing  over  a 


LESS,  v.]       EXERCISES  AND  PROBLEMS.  85 

pulley  without  friction,  along  a  smooth  horizontal  plane.     Find 
the  space  described  by  either  body  in  4  seconds. 

164.  If  a  body  weighing  20  kilogrammes  be  placed  on  a  plane 
which  is  made  to  descend  vertically  with  an  acceleration  of  one 
metre  per  second,  find  the  pressure  on  the  plane. 

165.  If  a  mass  of  m  units  be  placed  on  a  plane  which  is  made 
to  ascend  vertically  with  an  acceleration  /,  find  the  pressure  on 
the  plane. 

166.  A  stone,  weighing  P  Ibs.,  is  whirled  round  horizontally  by 
a  cord  I  feet  long,  having  one  end  fixed ;  find  the  time  of  revolution 
when  the  tension  of  the  cord  is  Q  Ibs. 


LESSON  Y.  —  Statics  of  a  Particle.  Parallel  Forces.  Mo- 
ments. Equilibrium  of  a  Rigid  Body.  The  Mechanical 
Powers. 

'  [In  Exercises  on  the  Mechanical  Powers,  the  chief  forces  concerned 
are,  a  weight  to  be  supported  (or  a  pressure  to  be  exerted),  and  a  power 
by  which  the  weight  is  supported  (or  the  pressure  exerted) ;  these  forces 
are  in  general  denoted  by  the  letters  W  and  P  respectively.  Unless 
otherwise  stated,  friction  and  the  stiffness  of  cords  are  neglected,  and 
lovers  are  supposed  to  be  straight,  horizontal,  perfectly  rigid,  without 
weight,  and  acted  upon  by  parallel  forces.] 

167.  Show  how  a  straight  line  is  employed  to  denote 
the  magnitude,  direction,  and  point  of  application  of  a 
force. 

168.  If  a  force  of  P  units  be  represented  by  a  straight 
line  a  inches  long,  what  force  will  a  straight  line  /  inches 
long  represent  ? 

169.  State  and  illustrate  the  simple  statical  Principles 
or  Axioms  which  may  be  designated  as  follows  :  — 

I.   Equal  Forces. 
II.   Action  and  Reaction. 

III.  Superposition  of  Forces. 

IV.  Transmissibility  of  Force. 
V.    Tension  of  a  Cord. 

170.  Distinguish  between  the  direction  and  the  line  of 
action  of  a  force. 

171.  Show  that  the  resultant  of  any  number  of  forces 


86  ELEMENTARY  PHYSIOS.  [CHAP.  i. 

acting  along  one  line  is  equal  to  the  algebraic  sum  of  the 
forces.  What,  then,  is  the  condition  of  equilibrium  for 
any  number  of  such  forces  ? 

172.  When  two  forces  act  in  the  same  direction  they 
have  a  resultant  of  12  Ibs.;  when  they  act  in  opposite 
directions  their  resultant  is  2  Ibs.     Find  the  forces. 

173.  Can  three  forces,  represented  by  the  numbers  5,  6, 
and  12,  keep  a  point  at  rest  ? 

'  174.  Enunciate  the  Parallelogram  of  Forces,  the  Tri- 
angle of  Forces,  and  the  Polygon  of  Forces ;  and  show 
that  all  three  propositions  are  deductions  from  the  corre- 
sponding propositions  relating  to  velocities. 

175.  Find  the  resultant  of  two  forces  of  8  Ibs.  and  15 
Ibs.  respectively,  acting  on  a  particle  at  right  angles  to  each 
other. 

176.  What  must  be  the  directions  of  two  forces  in  order 
that  their  resultant  may  be  the  greatest  possible,  and  also 
the  least  possible  ? 

177.  Show  that  as  the  angle  between  two  forces  is  in- 
creased their  resultant  will  be  diminished. 

178.  Show  how  to  find  the  resultant  of  two  forces  when 
the  directions  of  the  forces  and  of  the  resultant,  and  also 
the  magnitude  of  one  of  the  forces,  are  given. 

179.  Three, ,r,qpes,  P  A,  Q  A,  R  A,  are  knotted  together 
at  the  poiiio  A  ;  P  A  is  attached  to  a  tree,  Q  A  and  R  A 
are  pulled  by  two  men  ;  given  the-  angle  Q  A  R  and  the 
force  exerted  by  each  man,  show  how  to  find  the  pressure 
on  the  tree. 

ISO.  Two  forces  which  act  at  right  angles  on  a  point 
are  in  the  ratio  9  :  40,  and  their  resultant  is  123  Ibs.;  find 
the  forces. 

181.  The  resultant  of  two  forces,  P  and  Q,  is  perpen- 
dicular to  P;  show  that  it  is  less  than  Q. 

182.  Resolve  1*752  Ibs.  into  two  equal  forces  at  right 
angles. 

183.  Three  equal  forces  act  on  a  point  ;  find  the  con- 
ditions of  equilibrium. 

184.  A  horse  tows  a  boat  along  a  canal,  the  tow-rope 


LESS,  v.]       EXERCISES  AND  PROBLEMS.  87 

making  an  angle  of  30°  with  the  course  of  the  boat.  If 
the  horse  pull  with  a  force  of  2000  Ibs.,  find  the  effective 
component. 

185.  Two  rafters,  making  an  angle  of  120°,  support  a 
chandelier  weighing  80  Ibs.;   what  will  be  the  pressure 
along  each  rafter  1 

186.  A  cord  is  attached  to  two  fixed  points,  A  and  J?, 
in  the  same  horizontal  line,  and  bears  a  ring  weighing  10 
Ibs.  at  (7,  so  that  A  C  B  is  a  right  angle  ;  find  the  tension 
of  the  cord. 

187.  Forces  of  2,  4,  6,  and  8  Ibs.  respectively  act  along 
the  straight  lines  drawn  from  the  centre  of  a  square  to  the 
angular  points  taken  in  order.     Find  the  magnitude  and 
direction  of  the  resultant. 

188.  Distinguish    between    like    and   unlike  parallel 
forces. 

189.  Prove   that   the   resultant   of  two   like  parallel 
forces  is  equal  to  their  sum,  and  is  in  the  parallel  line 
which  divides  the  distance  between  the  forces  into  parts 
inversely  as  their  magnitudes. 

190.  Prove  that  the  resultant  of  two  unlike  parallel 
forces  is  equal  to  their  difference,  and  acts  in  the  direction 
of  the  larger  force  along  a  line  which  lies  beyond  the 
larger  force,  reckoning  from  the  smaller  force,  and  the  dis- 
tances of  which,  from   the   lines  of  act""*11   of  the  two 
forces,  are  inversely  as  the  magnitudes  of  the  forces. 

191.  Two  men  carry  a  weight  of  152  Ibs.  between  them 
on  a  pole  resting  on  one  shoulder  of  each  ;  the  weight  is 
three  times  as  far  from  one  as  from  the  other.     Find  how 
much  weight  each  supports,  the  weight  of  the  pole  being 
neglected. 

192.  If  Pand  Q  denote  two  like  parallel  forces,  a  the  dis- 
tance apart  of  their  lines  of  action,  and  x  the  distance  apart 
of  the  lines  of  action  of  P  and  the  resultant,  prove  that 


P+  Q 

193.    Prove  that  if  any  line  whatsoever  be  drawn  so 
as  to  intersect  the  lines  of  action  of  two  parallel  forces  and 


88  ELEMENTARY  PHYSICS.  [CHAP.  i. 

their  resultant,  the  portions  included  between  the  lines  of 
action  of  the  forces  respectively  and  that  of  their  resultant 
will  be  to  each  other  inversely  as  the  forces. 

194.  Show  how  to  determine  by  construction  the  re- 
sultant of  any  number  of  parallel  forces.     Is  it  necessary 
that  the  parallel  forces  should  be  in  one  plane  ? 

195.  Define  the  centre  of  a  system  of  parallel  forces. 

196.  Show,  from  geometrical   considerations,  that  the 
less  the  difference  in  magnitude  between  two  unlike  par- 
allel forces,  the  more  remote   is  the  point  of  application 
of  their  resultant.     If  this  difference  is  nothing,  where  is 
the  point  of  application  of  the  resultant  ? 

197.  When  two  unlike  parallel  forces  are  equal  in  mag- 
nitude, their  resultant,  by  the  general  solution,  is  equal  to 
xero.     Show,  however,   that   two    such  forces  are  not  in 
equilibrium,  and  explain  what  effect  they  will  produce  on 
the  body  to  which  they  are  applied. 

198.  Define  a  couple,  its  arm,  its  moment,  positive  and 
negative  couples. 

199.  Define  the  moment  of  a  force  with  respect  to  a 
point,  the  moment  being  the  proper  numerical  measure 
of  the  effect  of  the  force  to  produce  rotation  around  the 
point.    Distinguish  between  positive  and  negative  moments. 

200.  Define  the  moment  of  a  force  with  respect  to  a 
'line  or  axis. 

201.  Prove  that  the  algebraic   sum  of  the  moments  of 
three  parallel  forces  in  equilibrium  round  any  point  in  the 
same,  plane  is  equal  to  zero. 

202.  Let  Pv  P2,  P3,  &c.,  denote  any  number  of  parallel 
forces  acting  on  a  rigid  body  in  one  plane,  av  «2,  az,  &c., 
the  distances  of  their  lines  of  action  respectively  from  any 
point  of  reference  in  the  plane  ;  and  let  It,  denote  the  re- 
sultant of  the  forces,  x  its  distance  from  the  point  of  refer- 
ence :  prove  that 

R  =  P1  +  P2  +  P3  + +  Pn 

R  x  =  Pl  a1  +  P2  «2  +  P3  aB  +  .  .  .  .+Pn  an 

203.  What  are  the  two  conditions  of  equilibrium  of 


LESS,  v.]        EXERCISES  AND  PROBLEMS.  89 

any  number  of  parallel  forces  in  one  plane?  Examine 
the  cases  in  which  one  of  these  conditions  holds  true 
without  the  other. 

204.  Define  the  three  kinds  of  Levers.     To  which  kind 
do  the  follcTwing  objects  respectively  belong,  —  a  crow-bar, 
an  oar,  a  pump  handle,  a  common    balance,  the   common 
fire-tongs,  a  pair  of  nut-crackers,  a  pair   of  scissors,  thf 
treadle  of  a  lathe,  the  elbow-joint. 

205.  If,  in  a  lever,  P  and  W  are  the  two  parallel  forces, 
a  and  b  the  arms,  or  perpendicular  distances  from  the  ful' 
crum  to  the  lines  of  action  of  the  forces  respectively,  sho^Y 
that  the  condition  of  equilibrium  is  P  a   =    W  b. 

206.  If  R  denote  the  pressure  on  the  fulcrum  of  a  lever, 
find  R  for  each  of  the  three  kinds  of  levers. 

207.  Explain  what  is  meant  by  leverage,  or  mechanical 
advantage  in  general,  and  show  in  which  kind  of  lever  me- 
chanical advantage  is  always  gained,  and  in  which  kind  it 
is  always  lost. 

208.  If  weights  of  6  and  9  kilogrammes  balance  on  the 
ends  of  a  lever  10  metres  long,  find  the  position  of  the 
fulcrum. 

209.  On  a  lever  of  the  second  kind  a  weight  of  2^  Ibs. 
is  suspended  at  a  distance  of  8  inches  from  the  fulcrum  ; 
the  power  which  holds  it  in  equilibrium  is  4  Ibs.    Find  the 
length  of  the  lever. 

210.  A  lever  8  metres  long  is  supported  in  a  horizontal 
position  by  props  placed  at  its  extremities  ;  where  must  a 
weight  of  64  kilogrammes  be  hung  so  that  the  pressure 
on  one  of  the  props  shall  be  8  kilogrammes  ? 

211.  If  a  body  be  weighed  successively  in  the  two  pans 
of  a  false  balance   (i.  e.  a  balance  with  unequal   arms), 
prove  that  the  true  wreight  is  a  mean  proportional  between 
the  false  weights. 

212.  A  man  carries  a  bundle  at  the  end  of  a  stick  over 
his  shoulder  ;    as  the  portion  of  the  stick  between  his 
shoulder  and  his  hand  is  diminished,  show  that  the  press- 
ure on  his  shoulder  is  increased.     Does  this  change  alter 
his  pressure  on  the  ground  ? 


90  ELEMENTARY  PHYSICS.  [CHAP.  i. 

213.  At  points  5  inches  apart,  on  a  rod  20  inches  long, 
weights  of  1  lb.,  2  Ibs.,  3  Ibs.,  4  Ibs.,  and  5  Ibs.  are  sus- 

E  ended.    The  rod  is  to  be  supported  in  a  horizontal  position 
y  a  single  string  ;  at  what  point  must  the  string  be  tied  ? 

214.  An  iron  bar  8  feet  long  rests  with  its  extremities 
011  two  props,  A  and  B.     Weights  of  64  Ibs.  and  109  Ibs. 
are  suspended  at  points  distant  2  feet  and  5  feet  respec- 
tively from  A.     Find  the  pressure  on  each  prop. 

215.  Prove  the  condition  of  equilibrium  on  the  Wheel 
and  Axle,  viz.,  — 

P  x  radius  of  wheel  =  W  x  radius  of  axle. 

216.  Find  the  power  required  to  raise  1  ton  by  means 
of  a  wheel  and  axle,  in  which  the  radius  of  the  wheel  is 
50  times  that  of  the  axle. 

217.  Prove  the  condition  of  equilibrium  for  the  Single 
Movable  Pulley,  viz.   W  =  2  P. 

218.  Prove  the  condition  of  equilibrium  for  the  First 
System  of  Pulleys,  viz.   W  —  n  P,  in    which   n  denotes 
the  number  of  cords  at  the  lower  block. 

219.  Prove  the  condition  of  equilibrium  for  the  Second 
System  of  Pulleys,   viz.   W  —  2;*  P,  in  which  n  denotes 
the  number  of  movable  pulleys,  each  hanging  by  a  sepa- 
rate cord. 

220.  Prove  the  condition  of  equilibrium  for  the  Third 
System  of  Pulleys,  viz.  W  =  (2n— 1)  P,  in  which  n  de- 
notes  the  number  of  separate   cords,   each   cord   being 
attached  to  the  weight. 

221.  In  the  Second  System  of  Pulleys,   n  —  6,   and 
P  =  28  Ibs.;  find  W. 

222.  Find  the  mechanical  advantage  obtained  by  using 
the  Third  System  of  Pulleys  with  8  cords. 

223.  Prove  the  conditions  of  equilibrium  for  the  smooth 
Inclined  Plane,  viz.,  — 

1.  P  :  W  —  height  of  plane  :  length  of  plane, 

2.  R  :  W  —  base  of  plane  :  length  of  plane, 

in  which  R  denotes  the  reaction  of  the  plane,  and  P  is 
supposed  to  act  along  the  length  of  the  plane. 


LESS,  v.]       EXERCISES  AND  PROBLEMS.  91 

224.  The  base  of  an  inclined  plane  is  16  feet,  and  its 
height  is  12  feet  ;  find  the  weight  which  will  be  supported 
on  the  plane  by  a  power  of  10  Ibs. 

225.  Find  the  inclination  of  an  inclined  plane  if  the 
pressure  of  the  weight  on  the  plane  is  equal  to  the  power. 

226.  Prove  that  if  P  acts  horizontally  on  an  inclined 
plane,  P  :  I  V  =  height  of  plane  :  base  of  plane. 

227.  Describe  the  construction  of  a  Screiu,  and  show  its 
analogy  to  an  inclined  plane. 

228.  Prove  the  conditions  of  equilibrium  for  the  Screw, 
viz.,  — 

P    _  distance  between  two  threads 
W       circumference  described  by  P 

229.  If  5  turns  of  a  screw  carry  the  head  forward  1 
inch,  what  power  is  required  to  exert  a  pressure  of  1  ton, 
the  length  of  the  lever  being  2  feet. 

STATICS   OF   A   PARTICLE. 

230.  Let  P  and  Q  denote  two  forces,  R  their  resultant,  and  a,  0, 
0,  the  same  angles  as  in  Exercises  88  and  89  ;  solve  the  problems 
of  compounding  and  of  resolving  two  forces  by  simple  substitution 
in  the  results  of  Exercises  88  and  89,  and  prove  the  legitimacy  of 
this  method. 

231.  Taking  the  formulae  for  the  composition  of  two  forces,  ex- 
amine the  cases,  (1)  in  which  P  =  Q  and  a  =  120°,  (2)  in  which 
0  =  0°,  (3)  in  which  0  =  180°,  (4)  in  which  0  =  90°. 

232.  Show  that  the  effective  component  of  a  force  P  along  a 
line  which  makes  an  angle,  a  with  the  line  of  action  of  the  force  is 
equal  to  P  cos  a.     Examine  the  case  in  which  a  =  90°. 

233.  Investigate  the  analytic  method  of  compounding  any  num- 
ber of  forces  acting  on  a  point  in  one  plane,  and  establish  the  fol- 
lowing formulae,  — 

X  —  PI  cos  at  -f  P2  cos  a2  +  P3  cos  a8  +  ....+  P*  cos  a» 
Y  =  P!  sin  at  +  P2  sin  a2  +  P3  sin  a3  +....+  Pn  sin  a« 


234.  Prove  that  it  is  necessary  and  sufficient  for  the  equilibrium 
of  any  number  of  forces  applied  to  a  point  in  one  plane  that  the 
algebraic  sums  of  the  resolved  parts  of  the  forces  in  two  rectan- 
gular directions  in  the  plane  be  each  equal  to  zero. 


92  ELEMENTARY  PHYSICS.  [CHAP.  i. 

235.  Four  equal  forces,  each  equal  to  P,  act  on  a  point ;  the 
first  is  perpendicular  to  the  second,  the  resultant  of  the  first  two 
is  perpendicular  to  the  third,  and  the  resultant  of  the  first  three  is 
perpendicular  to  the  fourth.     Find  the  magnitude  of  the  resultant 
of  all  the  forces. 

236.  Four  forces,  equal  respectively  to  1  lb.,  2  Ibs.,  3  Ibs .,  and 
4  Ibs. ,  act  on  a  point ;  the  angle  between  the  directions  of  the  first 
and  third  is  a  right  angle,  and  likewise  the  angle  between  the  di- 
rections of  the  second  and  fourth  ;  while  the  angle  between  the 
directions  of  the  first  and  second  is  60°.     Find  the  magnitude  and 
direction  of  the  resultant. 

237.  Explain  how  the  force  of  the  current  may  be  employed  to 
urge  a  ferry-boat  across  a  river,  the  centre  of  the  boat  being  at- 
tached, by  means  of  a  long  rope,  to  a  mooring  in  the  middle  of  the 
stream. 

238.  What  force  must  a  man  exert  horizontally  to  push  a  weight 
of  336  Ibs.  a  distance  of  4  feet  from  the  vertical,  supposing  it  to 
be  suspended  from  a  hook  by  a  rope  20  feet  long  ? 

239.  The  ends  of  a  cord  6  metres  long  are  tied  to  two  hooks  in 
the  same  horizontal  line  3  metres  apart,  and  a  smooth  ring  sliding 
on  the  cord  sustains  a  weight  of  10  kilogrammes  ;  find  the  tension 
of  the  cord. 

240.  From  the  highest  point,  (7,  common  to  two  equal  rafters, 
C  A  and  C  B,  is  suspended  a  chandelier,  the  weight  of  which  is 
W  ;  find  the  horizontal  thrust  at  A  or  B. 

241.  A  man  with  a  perfectly  smooth  spherical  head  wears  a 
conical  hat ;  find  the  total  pressure  on  his  head.    What  remarkable 
consequence  follows,  if  we  suppose  the  hat  to  be  very  high,  its 
weight  remaining  the  same  ? 

242.  Find  the  resultant  of  three  forces  acting  on  a  point  in  lines 
at  right  angles  to  one  another. 

243.  Forces  act  on  a  point  in  any  direction  ;  find  the  magnitude 
and  direction  of  their  resultant. 

244.  What  are  the  conditions  of  equilibrium  of  a  particle  under 
the  action  of  any  system  of  forces  ? 

MOMENTS  AND   COUPLES. 

245.  Prove  that  if  any  number  of  forces  act  on  a  point  in  one 
plane  the  algebraic  sum  of  their  moments  about  any  point  in  their 
plane  is  equal  to  the  moment  of  their  resultant  about  the  same 
point      In  what  two  cases  is  this  algebraic  sum  equal  to  zero  ? 

246.  Define  the  axis  of  a  couple. 

247.  Show  that  a  couple  cannot  be  balanced  by  a  single  force, 
but  that  it  can  be  balanced  by  another  couple. 


LESS,  v.]        EXERCISES  AND  PROBLEMS.  93 

248.  Show  that  the  effect  of  a  force  at  a  point  not  in  its  line  of 
action  is  equivalent  to  an  equal  force  applied  at  that  point,  tor 
gether  with  a  couple  formed  of  the  original  force  and  an  equal 
force  supposed  to  be  applied  to  the  point  in  the  opposite  direction. 

249.  Prove  that  three  forces  which  act  in  consecutive  directions 
round  a  triangle,  and  are  represented  respectively  by  its  sides,  are 
not  in  equilibrium,  but  are  equivalent  to  a  couple. 

250.  Prove  that  any  two  couples  in  the  same  plane,  if  they  have 
equal  moments  with  unlike  signs,  balance  one  another. 

EQUILIBRIUM   OF   FORCES   IN   ONE   PLANE. 

251.  Show  that  any  system  of  forces  acting  in  one  plane  on  a 
rigid  body  can  be  reduced  to  a  single  force  or  to  a  couple. 

252.  Demonstrate  the  necessary  and   sufficient  conditions  of 
equilibrium  of  any  number  of  forces  acting  in  one  plane  on  a  rigid 
body,  viz.,  — 

1°.  The  algebraic  sums  of  the  projections  of  the  forces,  estimated 
along  any  two  rectangular  axes  in  the  same  plane  as  the  forces, 
must  each  be  equal  to  zero. 

2°.  The  algebraic  sum  of  the  moments  of  the  forces  with  respect 
to  any  point  in  the  plane  of  the  forces  must  be  equal  to  zero. 

253.  At  what  point  of  a  tree  must  a  rope  of  a  given  length  (I) 
be  attached,  in  order  that  a  man  on  the  ground,  pulling  at  the  other 
end,  may  produce  the  greatest  effect  in  overturning  the  tree  ? 

254.  A  sphere,  the  weight  of  which  is  W,  rests  on  two  planes 
inclined  at  angles  a  and  p  to  the  horizon  ;  find  the  normal  pressures 
on  the  planes. 

255.  A  hemispherical  bowl  in  a  fixed  position,  with  its  rim 
horizontal,  contains  a  weight  (  W )  which  is  attached  to  a  weight 
(P)  outside  the  bowl  by  means  of  a  string  passing  over  a  pulley 
on  the  rim  of  the  bowl.     Find  the  position  of  equilibrium  of  the 
weight  in  the  bowl,  friction  being  neglected. 

[In  the  next  four  Exercises  the  weight  of  a  material  body  is  to  be  con- 
sidered as  another  force  applied  at  the  centre  of  gravity  of  the  body,  and 
acting  vertically  downwards  (see  Lesson  VII.).] 

256.  A  roof  is  composed  of  equal  beams,  joined  in  pairs,  form- 
ing the  sides  of  isosceles  triangles ;  find  the  horizontal  thrust  on 
the  side  walls. 

257.  Two  beams  of  known  lengths,  m  and  n,  connected  together 
at  a  given  angle  0,  turn  about  a  horizontal  axis  at  their  point  of 
junction  ;  find  the  position  of  equilibrium  which  they  will  take  by 
tlreir  own  weight. 

258.  A  uniform  beam  rests  upon  a  rail,  with  one  extremity  in 


94  ELEMENTARY  PHYSICS.  [CHAP.  i. 

contact  with  a  smooth  vertical  wall ;  find  the  conditions  of  equi- 
librium. 

259.  A  ladder  rests  on  a  smooth  floor  with  its  upper  end  lean- 
ing against  a  smooth  vertical  wall;  determine  the  least  horizontal 
force  which,  applied  at  the  lower  end  of  the  ladder,  will  prevent 
it  from  slipping.     Find  also  how  this  force  is  altered  as  a  man  as- 
cends the  ladder. 

THE   MECHANICAL   POWERS. 

260.  The  weight  (  W)  being  given  in  each  of  the  three  kinds  of 
levers,  show  within  what  limits  the  magnitude  of  the  pressure  on 
the  fulcrum  will  lie. 

261.  Find  the  condition  of  equilibrium  in  a  combination  of 
levers  or  compound  lever. 

262.  A  bent  lever  has  equal  arms,  making  an  angle  of  120°  ;  find 
the  ratio  of  the  weights  at  the  ends  of  the  arms  when  the  lever  is 
in  equilibrium,  with  one  arm  horizontal. 

263.  Prove  that  in  a  combination  of  Wheels  and  Axles, 

P   _    product  of  radii  of  the  axles 
W  ~  product  of  radii  of  the  wheels 

264.  A  Wheel  and  Axle  is  applied  to  sustain  a  weight  on  an  in- 
clined plane,  the  string  being  parallel  to  the  plane  ;  find  the  con- 
ditions of  equilibrium. 

265.  Prove  that  the  mechanical  advantage  of  the  single  mov- 
able pulley  will  be  diminished  by  taking  into  account  the  weight 
of  the  pulley. 

266.  Find,  by  the  aid  of  trigonometry,  the  conditions  of  equi- 
librium on  the  Inclined  Plane,  viz.,  — 

P  =  W  sin  a ;  R  =  W  cos  a  : 

and  show  that  these  conditions  are  identical  with  those  given  in 
Exercise  223. 

267.  If  the  force  required  to  draw  a  wagon  on  a  horizontal  road 
be  -£Q  of  the  weight  of  the  wagon,  what  will  be  the  force  required 
to  draw  it  up  a  hill,  the  slope  of  which  is  1  in  40  ? 

268.  Two  unequal  weights  are  placed  on  two  smooth  inclined 
planes  having  a  common  height,  and  are  connected  by  a  fine  string 
passing  over  the  intersection  of  the  planes  ;  find  the  ratio  between 
the  weights  when  there  is  equilibrium. 

269.  The  length  of  the  power-arm  in  a  Screw  is  15  inches  ;  find 
the  distance  between  two  threads  of  the  Screw  in  order  that  the 
mechanical  advantage  may  be  30. 

270.  The  pitch  of  a  screw  is  30°,  the  radins  of  the  cylinder. 9 
inches,  and  the  length  of  the  arm  4  feet ;  find  the  power  that  will 
exert  a  pressure  of  1  ton. 


LESS,  vi.]      EXERCISES  AND  PROBLEMS.  95 

LESSON  VI.  —  Third  Law  of  Motion.    Impact. 

271.  State  the  Third  Law  of  Motion  in  the  terms  em- 
ployed by  Newton,  and  give  his  own  illustrations. 

272.  Distinguish  between  exterior  and  interior  forces, 
and  illustrate  the  distinction  by  an  example. 

273.  What  illustration  of  the  Third  Law  is  afforded  in 
leaping  from  a  small  boat  to  the  shore  1 

274.  If  a  nail  is  to  be  driven  into  a  board  which  is  not 
firmly  supported,  it  enters  much  better  provided  we  place 
behind  the  board  some  solid  body,  as  a  block  of  iron. 
Explain  this. 

275.  A  boat  is  100  yards  distant  from  a  ship,  and  a  man 
in  the  boat  hauls  the  boat  to  the  ship's  side  by  means  of 
a  rope  extending  from  the  ship's  side  to  the  boat.     The 
masses  of  the  boat  and  ship  are  200  Ibs.  and  1600  tons  re- 
spectively.    Find  where  the  ship  and  boat  meet,  supposing 
no  difference  in  the  resistance  of  the  water. 

276.  A   gun  weighing  5  tons  is  charged  with  a  ball 
weighing  28  Ibs. ;  if  the  gun  be  free  to  move,  with  what 
velocity  will  it  recoil  when  the  ball  leaves  it  with  a  ve- 
locity of  1000  feet  per  second  1 

277.  It  is  related  of  a  gentleman  that  he  thought  he 
had  found  the  means  of  commanding  at  any  time  a  fair 
wind  for  his  pleasure-boat  by  placing  an  immense  bellows 
in  the  stern,  blowing  against  the  sail.     Point  out  the  use- 
lessness  of  such  an  arrangement. 

278.  A  man  is  placed  on  a  perfectly  smooth  table ; 
show  how  he  may  get  off. 

279.  Distinguish   between   impulsive    and    continuous 
forces,  and  give  illustrations.     What  is  the  measure  of  an 
impulsive  force  ? 

280.  Distinguish  between  perfectly  elastic,  imperfectly 
elastic,  and  perfectly  inelastic  bodies.     Define  the  coefficient 
of  restitution. 

281.  State  the  principle  of  the  Conservation  of  Mo- 
mentum, and  explain  its  application  to  the  case  in  which 
two  equal  bodies  moving  with  equal  velocities  in  opposite 
directions  impinge  on  one  another. 


96  ELEMENTARY  PHYSICS.  [CHAP.  I. 

282.  Two  perfectly  inelastic  balls,  moving  along  the 
same  line  with  given  velocities,  impinge  directly  upon  each 
other  ;  find  the  common  velocity  after  impact. 

283.  An  inelastic  body,  weighing  250  kilogrammes  and 
moving  with  a  velocity  of  20  metres  per  second,  meets 
another  inelastic  body  weighing  300    kilogrammes   and 
moving  with  a  velocity  of  2  metres  per  second  in  the 
opposite  direction.  Find  the  common  velocity  after  impact. 

284.  Three  inelastic  balls,  weighing  respectively  5  Ibs., 
7  Ibs.,  and  8  Ibs.,  lie  in  the  same  straight  line.     The  first 
is  made  to  impinge  on  the  second  with  a  velocity  of  60  feet 
per  second  ;  the  first  and  second  together  then  impinge 
on  the  third.     Find  the  final  velocity. 

285.  What  fact  is   overlooked  in  the  following  objection  to 
the  Third  Law?     "When  an   obstacle  gives  way  under  a  force 
applied  to  it,  the  reaction  must  be  less  than  the  action  ;  for  how 
otherwise  can  the  yielding  of  the  obstacle  be  explained  than  by  the 
consideration  that  a  greater  force  overcomes  a  less  ? " 

288.  Illustrate  the  Third  Law  in  the  case  of  a  horse  and  cart, 
showing  what  force  or  forces  are  balanced  by  the  effort  of  the  horse, 
(1)  when  the  motion  is  uniform  ;  (2)  when  horse  and  cart  are  start- 
ing ;  (3)  when  the  cart  is  stopping,  the  horse  still  pulling;  (4)  when 
the  cart  is  stopping,  the  horse  being  backed. 

287.  Distinguish  between  impressed  and  effective  forces. 

288.  State  and  explain  D'Alembert's  Principle. 

289.  Two  heavy  bodies  are  connected  by  an  inextensible  string 
without  weight,  which  passes  over  a  fixed  smooth  pulley ;  deter- 
mine the  motion. 

290.  If  two  unequal  weights,  connected  by  a  string,  be  allowed  to 
fall,  the  string  being  vertical,  what  will  be  the  tension  of  the  string  ? 

291.  Determine  the  motions  of  two  weights,  W,  W ,  along  in- 
clined planes,  placed  back  to  back,  the  weights  being  connected  by 
a  thread. 

292.  Demonstrate  the  principle  of  the  Conservation  of  Momen- 
tum ,  stated  in  a  general  form  as  follows  :  the  sum  of  the  momenta 
of  the  parts  of  any  material  system,  estimated  in  any  direction,  is 
unchanged  by  the  action  of  interior  forces. 

293.  Demonstrate  the  formulae  for  the  impact  of  two  imper- 
fectly elastic  bodies,  viz. ,  — 

(m  +  n)  u'   =  m  u  +  n  v  -  e  n  (u  —  v), 
(m  +  n)  v'  —  m  u  4-  n  v  4-  e  m  (u  -  v)} 


LESS,  vi.]      EXERCISES  AND  PROBLEMS.  97 

in  which  e  is  the  coefficient  of  restitution,  m  and  n  the  masses  of  the 
bodies,  u  and  v  the  velocities  before  impact,  u1  and  v1  the  veloci- 
ties after  impact. 

294.  Show  that  the  formulae  of  the  preceding  Exercise  include 
the  cases  of  perfectly  elastic  and  perfectly  inelastic  bodies,  and 
adapt  the  formulae  to  these  cases. 

295.  Prove  that  the  relative  velocities  of  two  perfectly  elastic 
bodies  before  and  after  direct  impact  are  equal  and  opposite. 

296.  Define  vis  viva,  and  prove  that  in  the  impact  of  perfectly 
elastic  balls  no  vis  viva  is  lost. 

297.  Prove  that  in  the  impact  of  imperfectly  elastic  balls  vis 
viva  is  lost,  and  find  what  proportion  of  the  whole  vis  viva  is  lost. 

298.  A  ball  impinges  on  a  fixed  plane  with  a  velocity  given  in 
direction  and  magnitude  ;  the  value  of  the  coefficient  of  restitution 
between  the  ball  and  the  plane  is  e  ;  find  the  motion  of  the  ball 
after  impact,  friction  being  neglected. 

299.  A  billiard-ball  is  struck  from  one  corner.  A,  of  a  billiard- 
table,  A  BCD,  and  after  striking  three  of  its  sides  falls  into  the 
pocket  at  B  ;  show  that  the  alternate  sides  of  its  course  are  par- 
allel, and  find  the  distance  of  the  first  point  of  impact  from  B,  if 
A  B-a,  and  £(?=:&. 

300.  Review  the  evidence  in  favor  of  the  Third  Law  of  Motion. 


98  ELEMENTARY  PHYSICS.         [CHAP.  n. 


CHAPTER   II. 

THE  FORCES   OF  NATURE. 

LESSON  VII.  —  Universal  Gravitation. 

301.  When  a  pendulum  is  pulled  aside  from  its  vertical 
position  and  then  left  free,  why  does  it  not  swing  to  an 
equal  distance  on  the  other  side  of  the  vertical  position  ? 

302.  Newton,  in  his  pendulum  experiments,  employed 
as  the  bob  of  his   pendulum  a  box  into  which  he  put 
different  substances  equal  in  weight.     What  reason  was 
there  for  employing  the  box  'I 

303.  Comment  on  the  sentence  on  page  42    of  Mr. 
Stewart's  book,  beginning,  "  But  had  the  masses   been 
different  in  Newton's  experiment."     (See  top  of  page  14.) 

304.  What  is  the  angle  between  the  directions  of  plumb- 
lines  at  the  north  pole  and  at  the  equator  ? 

305.  State  the  proposition  referred  to  on  page  44  of 
Mr.    Stewart's  book  as   "a  well-known  proposition  in 
geometry." 

306.  State  Newton's  Law  of  Universal  Gravitation  in 
the  form  of  an  algebraic  formula,  and  derive  from  the 
formula  the  simplest  unit  of  attractive  force. 

307.  The  attractive  force  between  two  masses  is  4,  and 
the  values  of  the  masses  are  20  and  5  ;  find  their  distance 
apart. 

308.  Divide  a  given  mass  into  two  parts,  such  that  the 
mutual  attraction  of  the  parts  may  be  a  maximum. 

309.  Explain  precisely  how  far  the  Law  of  Gravitation  is  estab- 
lished by  the  proof  given  in  §  37  of  Mr.  Stewart's  book. 

310.  State  Kepler's  Laws.     How  were  they  established  ? 

311.  The  areas  which  revolving  bodies  describe  by  radii  drawn 
to  an  immovable  centre  of  force  lie  in  the  same  immovable  planes 
and  are  proportional  to  the  times  in  which  they  are  described. 
(Newton.) 


LESS,  vii.]      EXERCISES  AND  PROBLEMS.  99 

312.  Every  body  that  moves  in  any  curved  line  described  in  a 
plane,  in  such  a  manner  that  a  radius  drawn  from  the  body  to  a 
point  in  the  plane,  either  fixed  or  in  a  state  of  uniform  rectilinear 
motion,  describes  about  the  point  areas  proportional  to  the  times, 
is  urged  by  a  centripetal  force  directed  to  that  point.     (Neu'ton.} 

313.  If  a  body  describe  areas  proportional  to  the  times  about 
another  body,  however  moved,  the  force  which  acts  on  the  first 
body  is  the  resultant  of  two  forces,  — a  central  force  towards  the 
second  body,  and  an  accelerating  force  common  to  both  bodies. 
(Newton. ) 

314.  The  velocity  of  a  body  attracted  towards  an  immovable 
centre,  in  spaces  void  of  resistance,  is  inversely  as  the  perpendicu- 
lar let  fall  from  that  centre  on  the  tangent  to  the  path.     ( New- 
ton. ) 

315.  The  path  of  a  body  being  an  ellipse,  and  the  centre  of 
force  being  at  one  of  the  foci,  it  is  required  to  find  the  law  of  the 
force .     (Newton. ) 

316.  Give  the  dynamical  interpretation  of  Kepler's  Laws,  and 
their  relations  to  the  Law  of  Universal  Gravitation. 

317.  Show  that  Kepler's  Third  Law,  taken  in  connection  with 
Newton's  Third  Law  of  Motion,  leads  to  the  conclusion  that  the 
mutual  attraction  between  the  sun  and  a  planet  must  be  directly 
proportional  to  the  mass  of  the  sun  as  well  as  to  that  of  the 
planet. 

318.  Two  bodies  attract  one  another  according  to  the  Law  of 
Gravitation  ;  determine  the  actual  character  of  the  motion. 

319.  Prove  that  the  accurate  statement  of  Kepler's  Third  Law  is 

rrz .  7^2  _        -R3     .       R'* 

~  M+m  '   M+m> 

in  which  M  denotes  the  mass  of  the  sun,  m  and  m'  the  masses  of 
any  two  planets,  T  and  T'  their  periods,  and  R  and  R'  their  mean 
distances  from  the  sun.  What  inference  may  be  drawn  as  to  the 
masses  of  the  planets  ? 

320.  Give  a  summary  of  the  evidence  in  favor  of  the  Law  of 
Universal  Gravitation. 

321.  Investigate  the  effect  of  the  centrifugal  force  due  to  the 
earth's  daily  rotation  upon  the  weight  of  a  body  at  the  equator. 

322.  Find  the  time  of  revolution  of  the  earth,  which  would 
cause  bodies  to  have  no  weight  at  the  equator. 

323.  Investigate  the  effect  of  centrifugal  force  at  any  place  upon 
the  weight  of  a  body  at  that  place. 

324.  "Explain  how  the  centrifugal  force  due  to  rotation  tends 
to  alter  the  form  of  the  rotating  body. 

325.  Explain  the  total  variation  which  is  known  to  exist  be- 
tween the  force  of  gravity  at  the  equator  and  at  the  pole. 


100  ELEMENTARY  PHYSICS.         [CHAP.  n. 

LESSON  VIII. — Attwood's  Machine. 

326.  Why  is  it  that  a  ball  of  lead  and  another  of  cork, 
precisely  equal  in  volume,  will  not  fall  to  the  earth  with 
equal  velocities  1 

327.  In  Attwood's  Machine,  what  reason  is  there  for 
having  the  wheels  at  the  top  as  small  as  possible  ? 

328.  Solve  the  following  general  exercise  on  Attwood's 
Machine  :   Two  heavy  bodies  are  connected  by  an  inex- 
tensible  string,  which  passes  over  a  fixed  pulley  ;  find  the 
tension  of  the  string  and  the  motions  of  the  bodies. 

329.  In  Attwood's  Machine,  given  P  =  12  oz.,   Q  —  6 
oz. ;  find  the  pressure  on  the  axis  of  the  pulley. 

330.  In  Attwood's  Machine,  given  P  =  Q  —  18'6  oz.; 
what  weight  must  be  added  to  P  in  order  that  it  may 
descend  through  100  ft.  in  8  seconds  ? 

331.  In  Experiment  H  with  Attwood's  Machine  (see 
Stewart,  pp.  51,  52),  the  boxes  are  observed  to  rise  and 
fall  alternately  through  diminishing  distances  for   some 
time  ;  explain  this. 

332.  A  weight  of  two  pounds  hanging  vertically  draws  another 
weight  of  three  pounds  up  a  smooth  plane  inclined  at  an  angle  of 
30°  to  the  horizon  ;  find  the  space  described  in  4  seconds. 

333.  Two  scales  are  suspended  by  a  string  over  a  small  pulley  ; 
six  equal  bullets  are  placed  in  one  scale  and  six  in  the  other  ; 
show  that  the  tension  of  the  string  is  greater  with  this  arrange- 
ment of  the  bullets  than  with  any  other. 

334.  If  two  unequal  weights  connected  by  a  string  be  allowed 
to  fall,  the  string  being  vertical,  what  will  be  the  tension  of  the 
string  ? 

LESSON  IX.  —  Centre  of  Gravity.     Pendulum. 

335.  Give  an  exact  definition  of  the  centre  of  gravity 
of  a  body. 

336.  Define  and  illustrate  the  terms  homogeneous  body, 
plane  of  symmetry,  centre  of  figure  or  geometric  centre. 

337.  Where  is  the  centre  of  gravity  of  a  homogeneous 
body,  having  planes  of  symmetry  1 


LESS,  ix.]       EXERCISES  AND  PROBLEMS.  101 

Locate  the  centre  of  gravity  in  the  following  cases  :  A 
straight  line,  circumference  of  a  circle,  area  of  a  circle,  area 
of  a  parallelogram,  surface  of  a  sphere,  volume  of  a  sphere, 
convex  surface  of  a  right  cylinder.,  vdume  of  a  right  cylin- 
der, volume  of  a  parallelopiped. 

338.  Show  how  to  find  the  centre  'of  gravity  of 'any 
number  of  heavy  points.          .;  "  - 

339.  Weights  of  1,  2,  and  3  pounds  are  placed  alo^g 
the  same  line  a  foot  apart ;  find  their  centre  of  gravity. 

340.  Examine  the  equilibrium  of  a  body  suspended 
from  a  fixed  point. 

341.  Examine  the  equilibrium  of  a  body  resting  upon 
a  fixed  point. 

342.  Examine  the  equilibrium  of  a  body  resting  upon 
two  fixed  points. 

343.  Examine  the  equilibrium  of  a  body  resting  on 
three  or  more  fixed  points. 

344.  Examine  the  equilibrium  of  a  right  cylinder  rest- 
ing on  a  horizontal  table,  (1)  on  its  base,  (2)  on  its  side. 

345.  Explain  why  a  load  of  hay  will  overset  on  the 
side  of  a  hill,  when  a  load  of  iron  of  equal  weight  will 
pass  along  in  safety. 

346.  Explain  why  it  is  difficult,  if  not  impossible,  for 
a  person  standing  with  his  heels  against  a  wall,  to  pick  up 
a  cent  between  his  feet. 

347.  A  balance  is  in  equilibrium  with  horizontal  beam, 
unloaded  pans,  and  the  points  of  suspension  of  the  pans 
at  a  higher  level  than  the  fulcrum  ;   explain  the  effect 
produced  by  loading  the  pans. 

348.  Show  how  to  determine  the  value  of  the  accel- 
eration of  gravity  at  any  place  by  pendulum  observations. 

349.  Given  the  length  of   a  second's  pendulum,  find 
the  length  of  a  pendulum  which  will  oscillate  once  a 
minute. 

350.  If  a  pendulum  were  taken  to  a  place  where  the 
force  of  gravity  was  increased  fourfold,  how  much  would 
the  length  of  the  pendulum  have  to  be  changed  in  order 
that  the  time  of  oscillation  should  remain  the  same  as 
before  ? 


102  ELEMENTARY  PHYSICS.          [CHAP.  n. 

351.  A  pendulum  at  Paris  one  metre  long  was  found 
to  oscillate  in  1-00304  seconds  ;  find  the  value  of  g  at 
Paris. 

'  :*  352.   Sho\v  how  io  find  the  centre  of  gravity  of  a  triangle. 

&?3.  A  uniform  vfcar,  4^  feet  long,  weighs  10  Ibs.  ;  and  weights 
of  30  Ibs.  and  40  Ibs.  are^plaped  on  its  two  extremities  ;  on  what 
poiijc  -,\£il  ii;  bailee  ?-  t  f  ' 

'354:  Ho\v  long  a  piece'  mnsir  be  cut  off  from  one  end  of  a  rod 
"of  length  2  a,  in  order  that  the  centre  of  gravity  of  the  rod  may 
approach  towards  the  other  end  through  a  distance  b  ? 

355.  A  beetle  crawls  from  one  end  of  a  straight  fixed  rod  to 
the  other  end  ;  find  the  consequent  alteration  in  the  position  of 
the  centre  of  gravity  of  the  rod  and  beetle. 

356.  Two  homogeneous  spheres  of  equal  density  touch  each 
other  ;  find  the  distance  of  their  centre  of  gravity  from  the  point 
of  contact,  the  radii  being  respectively  8  inches  and  12  inches. 

357.  How  high  can  a  cylindrically  shaped  tower  of  r  metres 
radius  be  built  without  falling,  if  it  be  inclined  from  the  vertical 
by  an  angle  of  B  degrees  ? 

358.  Show  how  the  requisites  of  a  good  balance  may  be  satis- 
fied. 

359.  A  shopkeeper  has  correct  weights,  but  a  false  balance; 
supposing  that  he  serves  out  to  two  customers  articles  weighing 
W  Ibs.  by  his  balance,  using  first  one  scale  and  then  the  other, 
find  whether  he  gains  or  loses  on  the  whole,  and  how  much. 

360.  Prove  that  the  time  of  one  small  oscillation  of  a  simple 
pendulum  of  length  I  is  equal  to 


361.  A  pendulum,  which  would  oscillate  once  a  second  at  the 
equator,  would  gain  5  minutes  a  day  at  the  pole  ;  compare  equa- 
torial and  polar  gravity. 

362.  If  two  pendulums,  of  lengths  I  and  I',  at  different  points  on 
the  earth's  surface  make  in  the  same  time  numbers  of  vibrations 
which  are  in  the  ratio  m  :  m'  \  find  the  ratio  between  the  forces  of 
gravity  at  the  two  places. 

363.  Show  how  the  height  of  a  mountain  may  be  ascertained 
by  pendulum  observations. 

364.  A  seconds  pendulum  is  taken  to  the  top  of  a  mountain  of 
height  h  ;  find  the  number  of  seconds  it  will  lose  in  one  day. 

365.  A  seconds  pendulum,  on  being  taken  to  the  bottom  of  a 
mine,  was  found  to  lose  10  seconds  a  day  ;  find  the  depth  of  the 
mine,  given  that  the  earth's  radius  is  equal  to  4000  miles,  and 


LESS,  x.]        EXERCISES  AND  PROBLEMS.  103 

that  in  the  interior  of  the  earth  gravity  varies  directly  as  the  dis- 
tance from  the  earth's  centre. 

366.  A  pendulum,  when  taken  to  the  top  of  a  mountain,  is  ob- 
served to  lose  daily  just  twice  as  much  as  it  does  when  taken  to 
the  bottom  of  a  mine  in  the  neighborhood  ;  show  that  the  height 
of  the  mountain  is  equal  to  the  depth  of  the  mine. 

367.  The  times  of  oscillation  of  a  pendulum  are  observed  on 
the  earth's  surface  and  at  the  bottom  of  a  mine ;  hence  find  the 
radius  of  the  earth,  supposed  spherical. 


LESSON  X.  —  Forces  exhibited  in  Solids. 

368.  Define  and  illustrate  the  terms  coefficient  of  fric- 
tion^ angle  of  friction. 

369.  What  are  the  forces  acting  on  a  body  which  stands 
at  rest  on  the  side  of  a  hill  ? 

370.  What  are  the  forces  acting  on   a  ladder  which 
stands  with  one  end  on  a  rough  horizontal  floor  and  the 
other  end  resting  against  a  rough  vertical  wall  ] 

371.  Prove  that  the  coefficient  of  friction  is  equal  to 
the  tangent  of  the  angle  of  friction  (or  to  the  ratio  of  the 
height  to  the  base  of  an  inclined  plane  the  angle  of  in- 
clination of  which  is  equal  to  the  angle  of  friction). 

372.  A  body  will  just  rest  on  a  plane  inclined  at  an 
angle  of  45°  ;  find  the  coefficient  of  friction. 

373.  The  height  of  a  rough  inclined  plane  is  to  the 
length  as  3  :  5,  and  a  weight  of  30  Ibs.  is  just  supported 
on  the  plane  by  the  friction  ;  find  (1)  the  force  of  fric- 
tion, (2)  the  coefficient  of  friction. 

374.  Compare  the  strength  of  two  beams,  one  of  which 
is  twice  as  long  and  twice  as  deep  as  the  other,  their 
breadths  being  the  same. 

375.  Since,  of  beams  of  the  same  section,  the  deeper  is 
the  stronger,  why  are  not  beams  made  in  practice  exceed- 
ingly thin  and  deep  ? 


104  ELEMENTARY  PHYSICS.          [CHAP.  n. 

LESSON  XL  —  Forces  exhibited  in  Liquids. 

[In  the  exercises  on  this  Lesson  the  pressure  of  the  atmosphere  is 
neglected.    For  specific  gravities,  see  Appendix  V.] 

376.  Define  a  perfect  fluid.     Define  and  illustrate  vis- 
cosity. 

377.  State  the  chief  differences  between  a  solid  and  a 
fluid  ;  between  a  liquid  and  a  gas. 

378.  What  necessary  consequence  as  to  the  direction 
of  fluid  pressure  follows  immediately  from  the  definition 
of  a  perfect  fluid  ? 

379.  State  Pascal's  principle  in  a  mathematical  form. 
Why  cannot  the  principle  be  completely  established  by 
direct  experiment  ? 

380.  Show  that  any  force,  however  small,  may,  by 
transmission   through  a  fluid,  be  made  to   support   any 
weight,  however  large. 

381.  Show  how,  by  the  weight  of  a  few  ounces   of 
water,  the  strongest  cask  may  be  burst. 

382.  Prove  that  in  Bramah's  press  we  have  a  direct 
verification   of  the  general  principle  of  Mechanics,  that 
what  is  gained  in  force  is  lost  in  velocity. 

383.  A  vessel  full  of  liquid  has  two  pistons,  3  and  18 
centimetres  in  diameter  respectively  ;  what  pressure  on  the 
smaller  will  produce  a  pressure  of  900  kilogrammes  on 
the  larger  1 

384.  A  closed  vessel  full  of  liquid  has  a  weak  part  in 
its  upper  surface,  not  able  to  bear  a  pressure  greater  than 
9  Ibs.  per  square  foot.     If  a  piston  the  area  of  which  is 
one  square  inch  be  fitted  into  an  opening  in  the  upper 
surface,  what  pressure  applied  to  it  will  burst  the  ves- 
sel? 

385.  Prove  that  the  free  surface  of  a  liquid  at  rest  is 
normal  at  every  point    to    the   resultant  of  all   the  forces 
acting  at  that  point. 

386.  Explain  the  form  of  the  free  surface  of  a  liquid 
at  rest  on  the  surface  of  the  earth. 


LESS.  XL]     EXERCISES  AND  PROBLEMS.  105 

387.  If  we  consider  any  horizontal  plane  in  a  liquid 
at  rest  under  the  action  of  gravity,  the  pressure  on  the 
plane  is  proportional,  (1)  to  the  area  of  the  plane,  (2)  to 
the  depth  of  the  plane,  (3)  to  the  density  of  the  liquid. 
On  what  grounds  do  these  laws  rest  ] 

388.  Prove  that  the  pressure  on  any  horizontal  sur- 
face in  a  liquid  is  given  by  the  formula,  — 

.       (  area  (in  c.  m.2)  x  depth  (in  c.  m.) 
Pressure  (in  grammes)  =  j  x  d^nsity^ 

389.  Prove  that  the  pressure  on  any  vertical  surface 
in  a  liquid  is  given  by  the  formula 

P  =  SH  D 

where  P  denotes  the  pressure,  H  the  depth  of  the  centre 
of  gravity  of  the  surface  (supposed  uniform  in  density), 
and  D  the  density  of  the  liquid.  Specify  the  units  in 
which  P,  £,  H,  and  D  respectively  should  be  expressed. 
The  pressure  is  equal  to  the  weight  of  what  column  of 
water  1 

390.  Find  the  pressure  at  the  depth  of  30  metres  in  a 
lake. 

391.  What  height  must  a  column  of  water  have  which 
will   exert  a  pressure  of  1000  kilogrammes  per  square 
decimetre  ? 

392.  Required  the  pressure  on  a  rectangular  vertical 
side  of  a  tank  full  of  water,  the  height  of  the  tank  being 
4  metres,  and  its  breadth  being  80  centimetres. 

393.  Required  the  pressure  on  a  vertical  triangle  im- 
mersed in  water  with  the  base  in  the  surface,  the  base 
being  50  centimetres,  and  the  altitude  30  centimetres. 

394.  Prove  that  the  total  pressure  experienced  by  a  cu- 
bical vessel  full  of  water  is  equal  to  three  times  the  weight 
of  the  water. 

395.  Two  cubical  vessels,  the  edges  of  which  are  as 
two  to  .one,  are  filled  with  water  ;  compare  the  pressures, 
(1)  on  their  bases,  (2)  on  their  total  interior  surfaces. 

396.  Sketch  the  form  of  a  vessel  in  which  the  pressure 

5* 


106  ELEMENTARY  PHYSICS.          [CHAP.  n. 

against  the  sides  shall  much  exceed  the  pressure  on  the 
base. 

397.  Taking  the  density  of  mercury  as  13'6,  find  the 
total  interior  pressure  in  a  cylinder  full  of  mercury,  the 
height  of  the  cylinder  being  one  metre,  and  the  radius  of 
the  base  being  16  centimetres. 

39°  Prove  that  the  common  surface  of  two  liquids 
which  do  not  mix  must  be  horizontal. 

399.  Prove  that  the  heights  of  two  vertical  liquid  col- 
umns in  communication  are  inversely  as  the  densities  of 
the  liquids. 

400.  Distinguish  between  the  centre  of  gravity  and  the 
centre  of  pressure  of  a  vertical  surface  pressed  by  a  liquid. 

401.  How  will  the  pressure  on  the  base  of  a  vessel  con- 
taining water  be  affected  by  dipping  a  piece  of  metal  into 
the  water,  (1)  when  the  vessel  is  just  full  of  water,  (2) 
when  the  vessel  is  not  full  ? 

402.  Prove  that  when  a  body  is  placed  in  a  liquid,  it 
will  (1)  sink,  (2)  remain  at  rest,  or  (3)  rise  to  the  surface 
of  the  liquid,  according  as  its  density  (or  specific  gravity) 
is  (1)  greater  than,  (2)  equal  to,  or  (3)  less  than  that  of  the 
liquid. 

403.  Find  the  weight  of  a  boat  which  displaces  10  cu- 
bic metres  of  water. 

404.  Prove  that,  when  a  bod}7"  floats  on  the  surface  of 
a  liquid,  the  part  of  the  body  immersed  is  to  the  whole  body 
as  the  density  of  the  body  is  to  that  of  the  liquid. 

405.  Define  and  distinguish  between  density  and  spe- 
cific gravity,  and  explain  how  each  is  measured. 

406.  A  body  measuring  18  cubic  centimetres  floats  in 
water  with  its  whole  bulk  immersed  ;  find  its  weight. 

407.  Find  the  weight  in  water  of  100  cubic  centime- 
tres of  iron. 

408.  Find  the  weight  of  one  cubic  centimetre  of  a  body 
which  floats  in  water  with  one  fifth  of  its  volume  above 
the  surface. 

409.  An  irregularly  shaped  mass  of  granite   weighs 
182  kilogrammes  in  air  arid  117  kilogrammes  in  water  ; 
find  its  volume  and  its  specific  gravity. 


LESS.  XL]     EXERCISES  AND  PROBLEMS.  107 

410.  Two  bodies  differing  in  bulk  weigh  the  same  in 
water.     Which   will   weigh   the  most  in  mercury  and 
which  the  most  in  a  vacuum  1 

411.  A  man   (specific  gravity  1:12)  weighs   70  kilo- 
grammes.    What  volume  of  cork  will  be  required  to  just 
float  him  in  water  ? 

412.  A  cylinder  of  wood  floats  in  water  with  the^axis 
vertical.     How  much  will  it  be  depressed  by  putting  a 
weight  W  on  top  of  it  ? 

413.  Suppose  that  a  man  exerting  all  his  strength  can 
just  raise  a  weight  of  100  kilogrammes  ;  find  the  weight  of 
a  stone  (specific  gravity  2 '5)  which  he  can  raise  under 
water. 

414.  Show  how  to  find  the  specific  gravity  of  a  mix- 
ture of  gi\  en  volumes  of  any  number  of  given  substances. 

415.  Show  how  to  find  the  specific  gravity  of  a  mix- 
ture of  given  weights  of  any  number  of  given  substances. 

416.  Find  the  specific  gravity  of  a  mixture  of  equal 
volumes  of  water  and  alcohol,  supposing  no  contraction 
to  take  place. 

417.  Find  the  specific  gravity  of  a  mixture   of  equal 
weights  of  water  and  alcohol,  supposing  no  contraction 
to  take  place. 

418.  Three  liquids,  the  specific  gravities  of  which  are 
respectively  1*2,  0*96,  and  1*456,  are  mixed  in  the  pro- 
portions by  volume  of  18  parts  of  the  first  to  16  parts  of 
the  second  and  15  parts  of  the  third.     Find  the  specific 
gravity  of  the  mixture. 

419.  What  are  the  proportions  of  gold  and  silver  in  an 
alloy  of  these  two  metals  which  weighs  10  kilogrammes 
in  air  and  9*375  kilogrammes  in  water  ? 

420.  If  a  diamond  ring  weighs  69 '5  grammes  in  air 
and   64-5   grammes   in   water,   find   the    weight   of  the 
diamond  in  the  ring. 

421.  Find  the  specific  gravity  of  a  piece  of  lead  which 
weighs   47*48  grammes  in  air  and  43*33    grammes  in 
water. 

422.  Explain  a  method  of  finding  the  specific  gravity 
of  a  solid  lighter  than  water. 


108  ELEMENTARY  PHYSICS.          [CHAP.  n. 

423.  A  block  of  wood,  weighing  in  air  8  Ibs.,  is  tied  to 
a  piece  of  metal  weighing  6  Ibs.  ;  in  water  both  together 
weigh  4  Ibs.,  while  the  metal  alone  weighs  5  Ibs.    Find  the 
specific  gravity  of  the  wood. 

424.  A   crystal   of  salt   weighs   6 '3  grammes   in  air  ; 
when  covered  with  wax,  the  whole  weighs  8'22  grammes 
in  air,  and   3 '02   grammes   in   water ;   find   the   specific 
gravity  of  the  salt. 

425.  A  glass  ball,  weighing  10  grammes  in  air,  loses 
3.636  grammes  in  water,  and  2 -88  grammes  in  alcohol. 
What  is  the  specific  gravity  of  the  alcohol  ? 

426.  Prove  that  the  volumes  of  different  liquids  dis- 
placed  by  the  same  floating   body  are   inversely  as  the 
specific  gravities  of  the  liquids. 

How,  by  this  principle,  can  the  specific  gravity  of  a 
liquid  be  ascertained  ? 

427.  Prove  that  the  entire  interior  pressure  on  a  hollow  sphere 
full  of  a  liquid  is  equal  to  three  times  the  weight  of  the  liquid. 

428.  A  rectangle  is  just  immersed  vertically  in  a  liquid,  with 
one  side  in  the  surface  ;  divide  it  by  a  horizontal  line  into  two  parts 
which  shall  be  equally  pressed  by  the  liquid. 

429.  Divide  a  rectangle  just  immersed  vertically  in  a  liquid, 
with  one  side  in  the  surface,  by  horizontal  lines  into  n  parts  on 
which  the  pressures  shall  be  equal. 

430.  Distinguish  between  total  pressure  and  resultant  pressure 
by  the  aid  of  illustrations  (as  a  cubical  vessel  full  of  water,  a  tea- 
cup full  of  tea,  &c. ),  and  prove  that  when  a  vessel  of  any  shape  is 
tilled  with  a  liquid  the  resultant  pressure  is  equal  to  the  weight 
of  the  liquid. 

431.  A  hollow  cylinder  (height  h,  radius  of  base  r)  is  filled  with 
a  liquid  ;  compare  the  total  pressure  on  its  interior  surface  with 
the  weight  of  the  liquid. 

432.  A  right  cone  (altitude  h,  radius  of  base  r)  rests  on  its  base 
and  is  full  of  water;  compare  the  entire  interior  pressure  with  the 
weight  of  the  water. 

433.  Prove  that  the  centre  of  pressure  of  a  rectangular  vertical 
area  is  on  the  vertical  line  which  passes  through  the  centre  of 
gravity  of  the  area  at  a  distance  of  two  thirds  of  the  height  of  the 
area  from  the  surface  of  the  liquid. 

434.  Prove  that,  as  a  plane  area  is  lowered  vertically  in  a  liquid, 
the  centre  of  pressure  approaches,  and  ultimately  coincides  with, 
the  centre  of  gravity  of  the  area. 


LESS.  XL]       EXERCISES  AND  PROBLEMS.  109 

435.  State  the  general  conditions  of  equilibrium  of  a  floating 
body,  and  define  the  states  of  stable,  unstable,  and  neutral  equi- 
librium, with  illustrations. 

436.  Define  the  metacentre  of  a  floating  body  and  prove  that 
the  equilibrium  of  a  floating  body  is  stable,  unstable,  or  neutral, 
according  as  the  metacentre  is  above,  below,  or  at  the  centre  of 
^gravity  of  the  body. 

437.  A  ship  sailing  into  a  river  sinks  2  centimetres,  and  after 
discharging  12000  kilogrammes  of  her  cargo,  rises  1  centimetre  ; 
determine  the  weight  of  ship  and  cargo,  the  specific  gravity  of  sea- 
water  being  1.026. 

438.  What  quantities  of  zinc  and  copper  must  be  taken  to 
make  an  alloy  weighing  50  grammes,  and  the  specific  gravity  of 

-   which  shall  be  8-2 '( 

439.  Given  the  weights  and  specific  gravities  of  two  bodies  of 
different  kind  ;  find  the  specific  gravity  of  the  compound  formed 
by  mixing  them,  (1)  when  the  contraction  is  ~ih  part  of  the  sum 
of  the  component  volumes,  (2)  when  the  expansion  is  — th  part 
of  the  sum  of  the  component  volumes. 

440.  Required  the   specific  gravity  of  a  mixture   of  18  kilo- 
grammes of  sulphuric  acid  and  8  kilogrammes  of  water,  assuming 
that  the  contraction  is  -^. 

441.  Three  masses  of  gold,  silver,  and  a  compound  of  gold  and 
silver,  weigh  respectively,  P,  Q,  and  R  grammes  in  air,  and  p,  q, 
and  r  grammes  in  water;   find  the  weight  of  gold  in  the  com- 

, pound. 

442.  The  weight  W  of  a  vessel  full  of  air  (specific  gravity  p) 
and  the  weight  W  of  the  same  vessel  full  of  a  liquid   (specific 
gravity  <r)  being  given  ;  find  the  capacity  of  the  vessel. 

443.  Given  the  apparent  weight  P  of  a  body  in  air,  and  its 
specific  gravity  0"  ;  find  its  true  weight  (or  weight  in  vacuo). 

444.  A  cube  of  lead  (edge  4  centimetres)  is  to  be  sustained 
under  water  by  attaching  to  it  a  sphere  of  cork ;  required  the 
diameter  of  the  sphere  of  cork  which  will  just  sustain  it. 

445.  A  mass  of   copper  is   suspected  of  being  hollow.      Its 
weight  in  air  is  523  grammes,  in  water  447*5  grammes.     Find  the 
volume  of  the  interior  cavity. 

446.  A  piece  of  wood  (specific  gravity  0 '729)  is  in  the  form  of 
a  right  cone,  and  is  floating  in  water  with  its  axis  vertical  ;  deter- 
mine how  much  of  the  height  of  the  cone  will  be  submerged,  (1) 
when  the  vertex  is  below,  (2)  when  the  vertex  is  above. 

447.  An  iron  cone  is  floating  on  mercury  with  its  vertex  down- 
wards ;  required  the  ratio  of  the  altitude  of  the  cone  immersed 
to  the  total  altitude  of  the  cone. 


110  ELEMENTARY  PHYSICS.          [CHAR  n. 

448.  Determine  the  ratio  of  the  thickness  of  a  hollow  iron 
globe,  to  its  diameter  in  order  that  it  may  just  float  in  water. 

449.  A  sphere  of  cork  is  set  free  at  the  depth  of  80  feet  in  a 
lake  ;  in  what  time  will  it  reach  the  surface  of  the  lake,  supposing 
that  it  experiences  no  resistance  to  motion  from  the  water '( 

450.  Determine  the  general  relation  between  the  volumes  of 
liquids  displaced  by  a  hydrometer  of  variable  immersion  and  the 
specific  gravities  of  the  liquids. 

451.  Explain  the  graphical  method  of  graduating  a  hydrome- 
ter. 


LESSON  XII.  —  Forces  exhibited  in  Gases. 

[The  specific  gravities  of  gases  are  usually  referred  to  that  of  dry  air  at 
the  same  pressure  and  temperature  as  a  standard.  See  Appendix  V. , 
Table  of  Specific  Gravities.] 

452.  Describe  three  forms  of  expressing  the  pressure 
or  tension  of  a  gaseous  body  which  are  in  use. 

453.  Find  the  pressure  of  the  atmosphere  on  the  sur- 
face of  a  glass  globe,  20  centimetres  in  diameter,  when 
the  barometer  stands  at  76  centimetres. 

454.  The  body  of   a  man  of  average  size  exposes  a 
surface  of  about  1*5  square  metres  ;  find  the  total  atmos- 
pheric pressure  upon  it  when  the  barometer  stands  at  72 
centimetres. 

455.  What  is  the  pressure  of  the  atmosphere  per  unit 
of  area  in  dynamical  measure,  when  the  height  of  the 
barometer  is  h,  and  in  a  locality  where  the  force  of  grav- 
ity on  unit  of  mass  is  g  ? 

456.  Required  the  height  of  a  water  barometer  when 
the  mercury  barometer  stands  at  76  centimetres. 

457.  Find  the  height  of  an  alcohol  barometer,  when 
that  of  the  water  barometer  is  33  feet. 

458.  A  barometer  is  partly  filled  with  water  and  partly 
with  mercury,  the  height  of  the  water  being  three  times 
that  of  the  mercury  ;  find  the  total  height  when  the  press- 
ure of  the  atmosphere  is  1020  grammes  per  square  centi- 
metre. 

459.  A  barometric  tube  less  than  30  inches  in  length 


LESS,  xii.]     EXERCISES  AND  PROBLEMS.  Ill 

was  filled  with  mercury,  and  then  weighed.  It  was  then 
inverted  in  the  usual  way  over  a  basin  of  mercury,  at- 
tached to  the  beam  of  a  balance  so  as  to  be  in  a  vertical 
position,  and  again  weighed.  It  was  found  to  weigh  pre- 
cisely the  same  as  before.  Explain  this. 

460.  A  weight,  suspended  by  a   string   from  a  fixed 
point,  is  partially  immersed  in  water  ;  will  the  tension 
of  the  string  be  increased  or  diminished  as  the  barometer 
rises  ? 

461.  If  a  piece  of  glass  float  on  the  mercury  within  a 
barometer,  will  the  mercury  stand  higher  or  lower  in 
consequence  ? 

462.  Explain  how  depths  below  the  sea-level  may  be 
determined   by  means  of  barometric   observations   in   a 
diving-bell. 

463.  If  the  height  of  a  barometer  in  a  diving-bell  is 
200  inches,  the  height  at  the  sea-level  being  30  inches, 
find  the  depth  descended. 

464.  Prove  that  the  height  of  the  atmosphere  supposed 
homogeneous  (i.  e.  of  the  same  density  throughout  as  at 
the  surface  of  the  sea)  would  be  about  7987  metres,  or 
rather  less  than  5  miles. 

465.  Deduce   a  formula  for  finding   the   ascensional 
force  of  a  spherical  balloon. 

466.  Calculate   the   ascensional  force   of    a   spherical 
soap-bubble,  10  centimetres  in  diameter,  filled  with  coal- 
gas  at  a  tension  of   78  centimetres,  the  weight  of  the 
watery  film  being  neglected. 

467.  Prove  that  the  specific  gravity  of  a  gas  with  ref- 
erence to  water  is  equal  to  the  product  of  its  specific  grav- 
ity with  reference  to  air  and  the  specific  gravity  of  air 
with  reference  to  water. 

468.  Give  a  mathematical  statement  of  Boyle's  Law, 
supposed  absolutely  true.    Give  a  mathematical  statement 
of  this   law   which   includes  the   ascertained   deviations 
from  the  law. 

469.  Prove  that  the  specific  gravity  (and  also  the  mass 
of  a  given  volume)  of  a  gas  varies  directly  as  the  pressure 
applied  to  it. 


112  ELEMENTARY  PHYSICS.          [CHAP.  n. 

470.  A  cubic  foot  of  air  is  compressed  into  a  cubic 
inch  ;  find  the  pressure  required. 

471.  A  vessel  with  elastic  sides  contains  7*545  litres 
of  air  under  a  pressure  of  64  centimetres  ;  required  the 
volume  under  the  standard  pressure  of  76  centimetres. 

472.  If  a  small  hole  be  made  in  the  top  of  a  sub- 
merged diving-bell,  will  the  air  flow  out  or  the  water  flow 
in? 

473.  If  a  block  of  wood  be  floating  on  the  surface  of 
the  water  within  a  diving-bell,  how  will  it  be  affected  by 
the  descent  of  the  bell  1 

474.  How  would  the  tension  of  the  rope  which  sus- 
tains a  diving-bell  be  affected  by  opening  a  bottle  of  soda- 
water  in  the  bell  1 

475.  Suppose    a    cylindrical    diving-bell   is   lowered 
under  the  sea  by  means  of  a  rope  ;  show  that,  unless  air 
is  forced  in  from  above,  the  tension  of  the  rope  will  in- 
crease as  the  bell  descends. 

476.  Supposing  a  diving-bell  cylindrical,  and  that  no 
air  is  supplied  from  above  ;  find  the  height  to  which  the 
water  will  rise  in  the  bell  for  a  given  depth  descended. 

477.  If  a  cylindrical  tube  152  centimetres  long  be  half 
filled  with  mercury  and  then  inverted  over  mercury  in  a 
basin,  determine  how  high  the  mercury  will  stand  in  the 
tube  when  the  height  in  a  perfect  barometer  is  76  centi- 
metres. 

478.  Deduce  a  formula  for  expressing  the  rarefaction 
produced  in  an  air-pump  by  n  strokes  of  the  piston. 

479.  What  causes  prevent  us  from  obtaining  a  perfect 
vacuum  with  an  air-pump  1 

480.  Show  that  the  single-barrelled  air-pump  w^orks 
narder  as  the  rarefaction  proceeds. 

481.  If  the  capacity  of  the  receiver  of  an  air-pump  be 
three  times  that  of  the  barrel,  what  will  be  the  pressure 
of  the  air  in  the  receiver  after  three  strokes  of  the  piston, 
the  initial  pressure  being  15  pounds  per  square  inch  1 

482.  If  the  tension  of  the  air  in  the  receiver  of  an 
air-pump  is  reduced  to  one  fourth  its  original  amount  by 


LESS.  xii.J    EXERCISES  AND  PROBLEMS.  113 

three  strokes  of  the  piston,  compare  the  capacities  of  the 
receiver  and  barrel. 

483.  Find  the  force  required  to  raise  the  piston  of  the 
common  pump,  neglecting  friction  and  the  weight  of  the 
piston. 

484.  Prove  that  the  work  done  in  raising  a  given  quan- 
tity of  water  by  a  pump  is  the  same  as  would  be  done  in 
raising  the  water  through  an  equal  distance  in  buckets, 
friction  and  other  hurtful  resistances  being  neglected. 

485.  How  high  can  mercury  be  raised  by  a  common 
suction-pump  1 

486.  When  a  siphon  is  working,  what  would  be  the 
effect  of  making  a  small  hole  at  its  highest  point  1 

487.  What  effect  upon  the  action  of  a  siphon  would 
be  produced  by  carrying  it  up  a  mountain  1 

488.  What  effect  would  be  produced  on  the  action  of  a 
siphon,  if  the  atmosphere  were  suddenly  to  become  denser 
than  water? 

489.  The  common  radius  of  two  Magdeburg  hemispheres  is  16 
centimetres,  and  the  air  is  exhausted  until  the  interior  pressure  is 
reduced  to  4  centimetres  ;  required  the  force  necessary  to  separate 
the  hemispheres,  when  the  height  of  the  barometer  is  76  centi- 
metres. 

490.  Prove  that  the  weight  in  statical  or  gravitation  measure 
of  a  litre  of  dry  air  at  76  centimetres  pressure  in  any  locality 
is  proportional  to  the  intensity  of  the  force  of  gravity  at  that 
locality  ;  and  that  its  weight  in  dynamical  measure  is  propor- 
tional  to  the  square  of  the  intensity  of  the  force  of  gravity, 
temperature  being  supposed  constant. 

491.  Prove  that,  as  the  heights  above  the  earth's  surface  in- 
crease in  arithmetical  progression,  the  densities  of  the  air  decrease 
in  geometrical  progression,  temperature,  moisture,  and  intensity 
of  gravity  being  supposed  uniform. 

492.  Deduce  the  formulae  given  in  Appendix  V.  for  ascertaining 
the  li eight  of  a  mountain  by  barometric  observations. 

493.  Deduce  the  formulae  given  in  Appendix  V.  for  determining 
the  ascensional  force  of  a  balloon. 

494.  A  spherical  air-bubble  ascends  in  water  ;  given  its  diam- 
eter at  the  depth  a,  find  its  diameter  at  the  depth  ". 

495.  Explain  how  to  graduate  the  compressed-air  manometer, 
the  tube  being  supposed  uniform  in  diameter. 

H 


114  ELEMENTARY  PHYSICS.          [CHAP.  n. 

496.  A  small  quantity  of  air  is  left  in  the  upper  part  of  a 
barometric  tube  ;  determine  the  effect  on  the  height  of  the  mer- 
cury column. 

497.  The  weights  of  a  body  in  air  are  a,  a!,  corresponding  to 
the  heights  h,  h1,  of  the  barometer  ;  find  the  weight  corresponding 
to  a  height  h" . 

498.  Find  the  limit  to  the  exhaustion  of  air  by  an  air-pump 
due  to  the  fact  that  the  piston  does  not  descend  to  the  bottom  of 
the  barrel  (i.  e.  to  the  existence  of  what  may  be  termed  untrav- 
ersed  space). 

499.  Determine  the  law  of  the  rarefaction  of  air  in  an  air-pump, 
taking  into  account  the  existence  of  untraversed  space. 

500.  Find  the  law  of  the  increase  of  the  tension  of  the  air  in 
the  receiver  of  a  condensing-pump. 

501.  The  capacity  of  the  receiver  of  a  condensing-pump  is  10 
litres,  and  that  of  the  barrel  is  200  cubic  centimetres  ;  how  many 
strokes  are  required  to  bring  the  tension  of  the  air  in  the  receiver 
to  8  atmospheres  ? 

502.  Investigate  the  conditions  under  which  the  suction-pump 
will  not  work,  when  the  piston  does  not  descend  to  the  fixed 
valve. 


PAET  III. 
ANSWERS  AND  SOLUTIONS. 


I.— ANSWERS. 
LESSON  I. 

10.  5401440  square  feet.  11.  1240000  sq.  decimetres. 

12.  7415-375  cubic  yards.         13.  346768595  cubic  metres. 

14.  2000  litres.  15.  6  tonnes. 

16.  14-174765  grammes.  17.  As  1  to  1000000. 

18.  64  kilogrammes;  64grms.  20.  19200  kilogrammes. 

24.  1776  kilogrammes.  27.  3  miles  per  hour. 

28.  As  2  to  1.  29.  1527J  feet  per  second. 

30.  39  feet  per  second.  31.  13  feet  per  second. 

32.  As  Trtol.  33.  122522 -1114  kilogrammes. 

34.  714-28  cubic  centimetres.  35.  0'54. 

37.  The  density  of  every  substance  would  be  increased  - 
times. 

38.  The  density  of  the  wood  is  — rn-r  times  that  of  the 

2 

substance  whose  density  is  d. 

39.  -=£,  M  and  L  representing  in  general  a  mass  and  a 

length  respectively. 

41.    The  linear  velocity  must  be  equal  to  the  radius  of  the 
circle. 

42     -i-.-'-JL. 
21600'  1800' 

^;(2)-;    (8)L.  45.     360. 


116  ELEMENTARY  PHYSICS. 


LESSON  III. 

57.  (1)  The  direction  of  the  diagonal  of  a  parallelogram 
the  sides  of  which  are  4  in  the  direction  of  the  rowing,  and 
3  in  the  direction  of  the  current.  (2)  2  '5  miles.  (3)  1'5 
miles.  (4)  Half  an  hour.  (5)  Half  an  hour. 

63.  8  V  2  acting  towards  the  northwest. 

64.  The  ball  would  be  at  rest  relatively  to  the  ground. 

68.  (1)  36f  feet  per  second.     (2)  58f  feet  per  second  for 
th«  lirst  hour,  and  14§  feet  per  second  for  the  second  hour. 

69.  1624-/3  feet  per  second.  70.  5796  feet. 
71.  83'3  metres.           72.  490  metres.  73.  10  seconds. 
74.  (1)  402;5  feet ;  (2)  20  seconds.  76.  321  feet. 
77.  19 '6  metres  towards  the  ground. 

-83.  Height,  490  metres;  velocity,  98  metres  per  second. 

84.  95  '1  metres  per  second. 

87.  If  A  B  be  a  line  representing  in  magnitude  and  direc- 
tion the  current,  and  A  C  a  line  at  right  angles  to  the  first, 
representing  the  direction  of  the  boat's  motion  ;  then,  from 
B  with  radius  equal  to  the  velocity  of  the  rowing  describe  an 
arc  cutting  A  G  at  D ;  B  I)  is  the^direction  in  which  the  boat 
should  be  rowed. 

sin/3  sin  a 

89.  u  =  w  — — ~  ;  v  —  w  - . 

sin  <f>  sin  <f> 

90.  w  =  86-22  :  a  =  34°  15'  36"  :  £  =  19°  44'  24". 

91.  Each  velocity  =  7 '07  ;    each  angle  =  45°. 

93.  From  the   southwest,  with  a  velocity  of  4  V~2  miles 
per  hour. 

94.  He  should  aim  to  a  point  17*605  feet  in  advance  of  the 
deer,  or  along  a  line  making  an  angle  of  1°  40'  49''  with  the 
deer's  direction. 

95.  If  u  =  velocity  of  tube,  v  =  that  of  the   particle  ; 

then  —  =  sine  of  the  angle  of  inclination. 

99.  3600  g.     100.  Times,  2s  and  4s ;  velocities,  g  and  —  g. 
101.  87*662  metres.         102.  10  ^~g  feet  per  second. 

103.  1-2678.  104.    -  -f-  V/2  ga  feet  per  second. 

109.  Height  =  a. 


ANSWERS  AND  SOLUTIONS.  117 

In  exercises  on  projectiles,  let  v  denote  the  velocity  of  pro- 
jection, a  the  angle  between  the  direction  of  v  and  the  horizon. 

.    v2  sin2  a  _    2  v  sin  a 

114.  --  .  115.  -         —  . 

22«7  g 

116.  -  sin  2  a.  118.  7764Jeet. 

119.  If  u  and  v  denote  the  velocities,  and  a  /3  the  angles 
of  projection  respectively  ;  then,  distance  at  time  t  =  ^[u2  -+• 
v2  —  2  uv  cos  (a  —  j3)]  t. 

122.  —  4°  54'.  123.  233;238  feet. 

LESSON  IV. 

144.  739.  24  kilogrammes.  145.  (1)  480;  (2)  48  '98. 

146.  2  -3  nearly. 

147.  0-45359  m  ;  m;  0  '45359  m  ;  3219m; 
9-81  X  0  '45359m. 

148.  27  '734  feet. 

149.  The  rate  of  change  is  mass  X  acceleration  per  second. 

150.  772  800  000.       151.  10  '2s  ;  10  '2  times  the  first  force. 
157.  (1)  g  is  a  number  expressing  the  velocity  produced  in  a 

falling  body  in  unit  of  time. 

(2)  g  is  a  number  expressing  twice  the  distance  through 

which  a  body  falls  in  unit  of  time. 

(3)  g  is  a  number  expressing  the  weight  of  unit  of  mass 

in  dynamical  measure. 

160.  ^  Ibs.  161.  $  X  J  X  0*1  =  49  ;  v  =  |  0*. 

162.  s  =  -X^r;v  =  -gt.        163.  103-04  feet. 
m         2    '  mu 

164.  17-96  Ibs.  165.  pressure  =  m  (l  +  ^  J  . 

\  y  ' 

IP~T 

.  Time  of  revolution  =  2  TT  i  /  —  —  . 
V  Qg 


166. 


LESSON  Y. 

168.  P  f-.         172.  7  and  5.          173.  No.          175.  17. 

a 


118  ELEMENTARY  PHYSICS. 

• 

176.  The  same  and  the  opposite  directions  respectively. 

178.  Draw  from  any  point,  A,  three  lines  representing  the 
directions  of  the  two  forces  and  their  resultant,  and  take,  on 
the  line  which  represents  the  direction  of  the  given  force,  a 
length,  A  B,  to  represent  the  magnitude  of  this  force  ;  from 
B  draw  a  line  parallel  to  the  direction  of  the  other  force  till  it 
intersects  the  line  of  direction  of  the  resultant  at  C  ;  then  A  0 
is  the  magnitude  of  the  resultant. 

180.  27  and  120.  182.  Each  force  =  -7.12 . 

V/2 

183.  Their  lines  of  action  must  form  an  equilateral  triangle. 
185.  80  Ibs.  186.  5  V~21bs. 

187.  Magnitude  =  4  y  2  ;  direction  is  perpendicular  to  one 
of  the  sides  of  the  square. 
191.  38  and  114. 
206.  (1)P+Q,     (2)0  — P,     (3)P  —  Q. 

208.  4  metres  from  the  end  to  which  the  weight  of  9  kilo- 
grammes is  attached. 

209.  4  feet.  210.  7  m.  from  the  given  pressure. 

213.  6§  inches  from  the  end  supporting  the  weight  of  5  Ibs. 

214.  84J  and  88|  Ibs.  216.  40  Ibs. 
221.  1792  Ibs.              222.  255.            224.  16§  Ibs. 

OK 

225.45°.  229.^-.  235.  2  P. 

O  7T 

236.  6'89  Ibs.  ;  angle  of  R  with  first  force,  102°  16'. 

10 
238.  67  "2  Ibs.  239.  ~j^  kilogrammes. 

240.  Horizontal  thrust  =\W  cot  a,  where  a  =  inclina- 
tion of  either  rafter  to  the  horizon. 

241.  (1)  Pressure  =  JFcoseca,  where  W  =  weight  of  hat, 

a  =  semi-angle  of  the  cone. 
(2)  The  pressure  becomes  infinite. 

242.  If  P,  Q,  S,  are  the  forces,  £  their  resultant,  then,  — 


ANSWERS  AND  SOLUTIONS.  119 

243.  R*  =  (S  P  cos  a)2  +  (S  P  cos  /3)2  +  (S  P  cos  7)2 

S  P  cos  a                   S  P  cos  8                   S  P  cos  7 
cos  0 ,  cos  ^  =         ^       ,  cos  a; . 

244.  It  is  necessary  and  sufficient  for   the   equilibrium  of 
any  number  of  forces  applied  to  a  point,  that  the  sums  of  the 
resolved  parts  of  the  forces  in  the  directions  of  three  intersect- 
ing lines  not  in  one  plane  be  separately  equal  to  zero. 

253.  The  point  of  attachment  should  be  as  far  above  the 
ground  as  the  man  is  from  the  foot  of  the  tree. 

JFsin/3 

254.  Pressure   on   1st  plane  =  - — — . 

sin  (a  H-  /3) 

JPsina 

Pressure  on  2d  plane  =  — — —  . 

sin  (a  +  j8) 

255.  Let  0  =  angle  which  a  line  drawn  from  W  to  centre 
of  hemisphere  makes  with  the  horizon  ;  then,  — 

P  ±  y'  (8  W*  +  P2 

cos  4  *  =  \  w 

256.  Let  W  =  weight  of  each  beam,  0  =  its   angle  with 
the  horizon,  then,  horizontal  thrust  —  ^  W  cot  0. 

257.  Let  0  =  angle  of  m  with  horizon  ;  then,  — 

m2  -f-  n2  cos  0 


tan  0  =  - 


sin  0 


258.  Let  W  =  weight  of  the  beam,  P  =  pressure  of  beam 
on  the  rail,  R  =  pressure  of  beam  on  the  wall,  0  —  angle  of 
beam  with  the  wall,   a  =  half  the  length  of  the  beam,  b  = 
distance  of  the  rail  from  the  wall ;  then,  — 

;        p  =  W  cosec  6  ;        R  =  Wcot  0. 

259.  Let  W  denote  the  weight  of  the  ladder,  m  and  n  the 
segments  into  which  its  centre  of  gravity  divides  it,   0  the 
angle  which  the  ladder  makes  with  the  floor,  P  the  horizontal 
force  required  ;  then,  — 

P  =  — ^ —  W  cot  0. 


120  ELEMENTARY  PHYSICS. 

260.    (1)  Between  W  and  infinity. 

(2)  "        0  and  W. 

(3)  "        0  and  infinity. 

__  P    _  product  of  the  weight  arms 

W       product  of  the  power  arms  ' 
262.  As  1  to  2. 

264.  Let  P  denote  the  power  applied  to  the  wheel,  a  and  b 
radii  of  wheel  and  axle,  a  the  angle  of  the  inclined  plane 
with  the  horizon,  W  the  weight  on  the  plane  ;  then,  — 

p  _  W  b  sin  a 

a 
W 

267.  —  4-  T/O  of  the  pressure  on  the  plane. 

268.  The  weights  are  as  the  lengths  of  the  planes. 

269.  TT  inches.  270.  186 '86  Ibs. 


LESSON  VI. 


275.  sfjHta  °f  a  foot  from  the  i 
376.  2  '8  feet  per  second. 
278.  By  pushing  against  the  air  with  the  palms  of  his 
hands. 

282.  Let  m,  mf,  denote  the  masses,  v,  v1,  their  velocities, 
then,  — 

m  v  -f-  m1  vf 

velocity  alter  impact  =  —  - —  . 

m  4-  mf 

283.  8  metres  per  second.  284.  15  metres  per  second. 

m  —  mf  .2  m  m1 

289.  Acceleration  =  -          •  g  ;         tension  =  —    — ,  g. 

m  -\-  m'  rm-\-m 

290.  It  will  be  nothing. 

?7i  sin  a  —  m1  sin  at 

291.  Acceleration  =  —  — -. g  ; 

m  4-  mf 

m  mf  (sin  a  4~  sin  a') 

tension  of  the  string  —  —  —  g. 

m  4-  mf 


ANSWERS  AND  SOLUTIONS.  121 

298.  Let  u,  v,  denote  the  velocities  before  and  after  im- 
pact respectively,  it  v,  the  angles  of  incidence  and  reflection  ; 
then,  — 

tan  i  -—  e  tan  v, 

v2  =  V?  (sin2  i  +  e2  cos2  i). 

299.  §  b. 

LESSON  VII. 

304.  90°.         307.  5.         308.  The  parts  must  be  equal. 

321.  The  centrifugal  force  at  the  equator  diminishes  the 
weight  of  a  body  by  about  YTyth  part. 

322.  17  times  faster  than  it  now  revolves. 

323.  The    component    of    centrifugal    force    diminishing 

gravity  =  ~~~m^~  cos  2  X,  where  R  —  earth's  radius,  T  =  time 

of  revolution,  X  =  latitude. 

325.  The  total  variation  amounts  to  yirth  part  of  the 
weight  of  a  body.  In  other  words,  a  body  which  weighs 
194  Ibs.  on  the  equator  would  weigh  1951bs.,  very  nearly, 
at  the  pole.  Centrifugal  force  causes  a  part  of  this  difference, 
and  the  variation  in  the  earth's  attractive  force  due  to  its 
spheroidal  shape  produces  the  remainder  of  the  difference. 


LESSON  VIII. 

328.  Let  P  and  Q  be  the  weights  of  the  bodies,  T  the 
tension  of  the  string,  0  the  acceleration  ;  then,  — 


~  P+Q' 

329.  16  oz.  330.  4  Ibs. 

332.  £  g.  334.  Nothing. 

LESSON  IX. 

337.  At,  the  centre  of  figure  in  each  case. 
339.  At  a  point  8  inches  from  the  3  Ibs. 
6 


122  ELEMENTARY  PHYSICS. 

349.  3600  times  the  length  of  the  second's  pendulum. 

350.  Fourfold.  351.  9 '8098. 

353.  At  a  distance  of  21  inches  from  the  40  Ibs. 

354.  2ft. 

355. ,  where    I  =  length  of  rod,  a  —  its  weight, 

J  —  weight  of  beetle. 

356.  Let  D  =  density  of  the  sphere  whose  radius   is  8 
inches,  Df  =  density  of  the  other  sphere  ;  then  their  common 

540  D1 
centre  of  gravity  is  at  a  distance  of  —&f~rii  ^rom  t^ie 

o  U  ~T~  At  JJi 
8 -inch  sphere. 

357.  2  r  cosec  0.  359.  Weight  lost  =   W  ^~^. 

361.  As  144  to  145^|F.         362.  As  m2  I  to  ml*  I'. 

9  h 

364.  — — ,  h  being  expressed  in  feet. 

365.  0-94  of  a  mile. 

367.  Earth's  radius  =  -^ — — %,  where  tl  is  the  time  of  an 

^2  —  *i 
oscillation  at  the  surface,  and  t2  at  the  given  depth  h. 

LESSON  X. 

372.  1.  373.  18  Ibs.  ;  075. 

374.  Twice  as  strong. 

LESSON  XL 

383.  25  kilogrammes.  384.  1  oz. 

390.  3  kilogrammes  per  sq.  centimetre.     391.  100  metres. 
392.  6400  kilogrammes.  393.  7500  grammes. 

395.  As  8  :  1,  in  both  cases.        397.  348160  TT  grammes. 
403.  10000  kilogrammes.     406.  18  grms.     407.  725  grms. 
408.  0-8  gramme.  409.  2'8. 

410.  The  smaller,  in  mercury  ;  and  the  larger,  in  a  vacuum. 

W 

411.  9868  cubic  centimetres.  412.  — 5. 

TTT2 


ANSWERS  AND  SOLUTIONS.  123 

413.  166f  kilogrammes. 

414.  Let  Vit  V~z,  V-z,  &<?•,  be  the  given  volumes^  <rlt  <r2, 
(T3,  &c.,  the  specific  gravities,  <r'  the  specific  gravity  of  the 
mixture  ;  then,  — 

fff      -=      . '—  f 

415.  Let  Wi,  W<z,  W$,  &c.,  be  the  given  weights,  0*1,  <r2, 
0-3,  &e.,  the  specific  gravities,  <rf  the  specific  gravity  of  the 
mixture ;  then,  — 

Wi  +  W2  +  W3  +  &c. 

—1  H ^    _j 3    +  &c> 

416.  0-8975.  417.  0*8857.  418.  1*2. 

419.  7511  grammes  of  gold,  and  2489  grammes  of  silver. 

420.  3'5  grammes.      421.  11 '2.  423.  |. 
424.  1-9  nearly.           425.  0.792. 

428.  The  dividing  line  must  be  at  the  depth  -7— ,  where  h 

\/2 
denotes  the  vertical  edge. 

429.  Let  h  =  the  vertical  edge  ;  then  the  lines  of  division 
are  at  the  depths 


J 

\ 


-A,&c. 


Total  pressure 

Weight  of  the  liquid 

Total  pressure 

4;O4i.     — 


Weight  of  the  liquid  r 

437.  947076  tonnesv 

438.  Zinc  17 '82  grammes,  copper  32 '18  grammes. 

439.  If  W,Wi  and  o-,  <rf  denote  the  weights  and  specific 
gravities   respectively  of  the  two  bodies,   then  the  required 
specific  gravities  are,  — 


m  . 

\  >        _      •  *  ^  ' 


W 


124  ELEMENTARY  PHYSICS. 

440.  1-51  nearly.     441.  P.  Qr~  E-q~.     442.   W*  ~  W 
Qp  —  Pq  <r   ~p 

443.  If  cr'  =  specific  gravity  of  the  material  of  which  tlit 
weights  are  made,  p  =  specific  gravity  of  the  air  ;  then,  — 

true  weight  =  P  .  —  . • 

o-f     <j  — p 

444.  11-85  centimetres.  445.  16'1  square  centimetres. 

446.  (1)  0'9  of  the  height ;  (2)  0'647  of  the  height. 

447.  The  altitudes  of  the  two  cones  are  inversely  as  the 
cube  roots  of  the  specific  gravities  of  the  cone  and  the  liquid, 
whatever  be  the  vertex-angle  of  the  cone. 

448.  Thickness  :  diameter  =  £  (1  —  ^/a  —  1)  :  1. 

449.  1-252  seconds. 


LESSON  XII. 

453.  103-33  kilogrammes.  454.  14684  kilogrammes. 

455.  13  -596  #  h.  456.  1033  '3  centimetres. 

457.  41  -51  feet.  458.  245  centimetres. 

460.  Diminished.  461.  Neither. 

463.  187  feet  9  inches.  465.  See  Appendix  V. 

.cc    2000  TT 

466.  —  -  —  grammes,  nearly. 

o 

468.  If  we  have  a  volume  V  of  gas  at  the  pressure  7^,  and 
then  change  the  volume  to  V?  the  pressure  becoming  h1  ;  then 
Mariotte's  law  asserts  that,  — 

Vh 

V  :  V<  =  k>  :  h,    or  =  1. 


But  experiments  have  shown  that  gases,  except  hydrogen, 
are  more  compressible  than  the  law  indicates,  that  is,  for  all 

VJi 
gases  except  hydrogen  •          <^  1. 

YJi 
But  for  hydrogen      TFTT,  >  I- 


470.  1728  times  the  initial  pressure. 

471.  6  -354  litres.  472.  The  air  will  flow  out. 


ANSWERS  AND  SOLUTIONS.  125 

473.  More  of  it  will  be  submerged. 

474.  The  tension  will  be  diminished. 

476.  Let  h  =  depth  descended  in  feet,  Jc  =  height  of  a 
water  barometer,  in  feet ;  then,  — 

height  ascended  in  the  bell  _       h 
total  height  of  the  bell          h  +  k ' 

477.  46-968  centimetres. 

478.  Let   V  =  volume  of  the  receiver,  v  =  volume  of  the 
barrel,  D  =  initial  density  of  the  air  in  the  receiver,  Dn  = 
the  density  of  the  air  after  n  strokes  ;  then,  — 


481,  6fi  Ibs.  per  sq.  inch.  482.  As  5874  to  1. 

483.  It  is  equal  to  the  weight  of  a  column  of  water  having 
i  section  equal  to  the  area  of  the  piston,  and  a  height  equal 
to  the  height  of  the  column  of  water  raised. 

485.  Not  more  than  76  centimetres,  by  the-  suction  prin- 
ciple. 

436.  The  siphon  would  cease  working  and  the  water  in  the 
arms  would  flow  out. 

487.  The  rapidity  of  the  flow,  and  the  possible  height  of 
the  short  arm,  would  be  diminished. 

488.  The  siphon  would  work  backwards,  that  is,  the  flow 
would  be  from  the  long  arm  into,  and  out  of,  the  short  arm. 

489.  786'875  kilogrammes. 

494.  If  d  =  diameter  at  depth  a,  df  =  diameter  at  depth  -, 
h  =  height  of  a  water  barometer  ;  then,  — 


df  = 


496.  Let  I  =  length  of  the  tube,  h  =  height  of  a  perfect 
barometer,  y  =  length  of  the  air-column  in  the  tube  at  the 
pressure  h,  x  —  depression  produced  by  the  air  ;  then,  — 


126  ELEMENTARY  PHYSICS. 

497-  a  +  T=¥  (a'-a}- 

AQQ    Initial  density  of  the  air        capacity  of  barrel 
Final  density  attainable  ~  untraversed  space* 

499.  Let  V,  v,  denote  the  volumes  of  the  receiver  and 
barrel  respectively,  v1  the  untraversed  space,  D  the  initial 
density  of  the  air,  Dn  the  density  after  n  strokes  ;  then,  — 


500.  Let  A  and  B  be  the  capacities  of  the  receiver  and 
barrel  respectively,  h  the  measure  of  the  pressure  of  the  at- 
mosphere ;  then,  — 

tension  after  n  strokes  =  -  7-  —  .  h. 
A 

501.  350  strokes. 

502.  The   general   condition  is,    that  the  pump  will  not 
work  unless  the  play  of  the  piston  be  greater  than  the  square 
of  the  distance  from  the  surface  of  the  water  in  the  well  to  the 
highest  position  of  the  piston  divided  by  four  times  the  height 
of  the  water-barometer. 


ANSWERS  AND  SOLUTIONS.  127 

II.—  SOLUTIONS. 
LESSON  I. 

19.  The  unit  of  mass  in  the  Metric  System  (the  kilo- 
gramme), which  is,  strictly  speaking,  the  quantity  of  matter  in 
a  certain  platinum  weight  kept  in  Paris,  was  intended  to  be, 
and  may  "be  taken  as,  equal  to  the  mass  of  one  unit  of  volume 
(the  litre)  of  pure  water  at  4°  C. 

Hence,  2  litres  of  water  weigh  2  kilogrammes,  and  in  gen- 
eral a  litres  \veigh  a  kilogrammes. 

21.  The  density  of  a  body  is  the  ratio  of  its  mass  to  its  vol- 
ume, or,  in  symbols, 


D  denoting  density,  M  mass,  and  V  volume. 

The  numerical  measure  of  the  density  of  a  substance  is  ob- 
tained by  taking  the  unit  of  volume,  or  putting  V  =  1,  in 
which  case  D  =  M,  or  the  measure  of  density  is  the  number 
of  units  of  mass  in  unit  of  volume  of  the  substance. 

The  value  of  the  density  of  a  substance  evidently  depends 
on  the  units  of  mass  and  volume  adopted,  but  density  is 
always  the  ratio  of  a  mass  to  the  cube  of  a  length,  or,  as  it 

has  been  expressed,  the  dimensions  of  density  are  -^  ,  M  de- 

noting a  mass,  and  L  a  length. 

23.  Because  the  unit  of  mass  in  the  Metric  System  is  the 
~mass  of  unit  of  volume  of  pure  water  at  4°  C.  ;  or,  in  other 
words,  because,  in  the  case  of  water,  when  V  —  1,  M  =  1, 

and  therefore,  — 


26.  The  word  weighty  both  in  scientific  and  common  lan- 
guage, is  usually  employed  to  denote  mass  or  quantity  of 
matter.  But  the  weight  of  a  body,  properly  speaking,  is  the 
measure  of  the  force  with  which  the  body  is  drawn  towards  the 
centre  of  the  earth.  This  force  is  different  in  different  places, 


128  ELEMENTARY  PHYSICS. 

being  greater,  for  example,  at  the  pole  than  at  the  equator, 

and  greater  at  the  level  of  the  sea  than  at  the  top  of  a  moun- 
v  tain.  The  mass  of  a  body  is  the  quantity  of  matter  which 
^  it  contains;  this  is  invariable,  and  would  remain  the  same 

even  if  the  force   of  gravity  did   not   exist,  in  which   case 

bodies  would  have  no  weight. 

LESSON  III. 

Representation  of  Velocities  and  Forces.  —  A  velocity  (or  a 
force)  is  said  to  be  represented  geometrically  by  drawing  a 
straight  line  in  the  direction  of  the  given  velocity  (or  in  the 
direction  in  which  the  given  force  acts)  and  making  the  line 
as  many  units  long  as  there  are  units  in  the  given  velocity  (or 
in  the  given  force).  For  example,  a  velocity  of  10  feet  per 
second  towards  the  north  would  be  represented  on  paper  by  a 
line  10  inches  long,  drawn  as  in  maps,  &c.,  perpendicular  to 
the  upper  edge  of  the  paper.  A  force  of  16  units  acting  toward 
the  east  might  be  represented  on  paper  of  moderate  size  by  a 
line  4  inches  long,  taking  for  convenience  one  quarter  of  an 
inch  to  represent  the  unit  of  force,  the  line  being  drawn  from 
a  point  where  the  force  is  supposed  to  act,  towards  the  right- 
hand  edge  of  the  paper  and  perpendicular  to  that  edge.  An 
arrow -head  may  be  added,  as  in  Fig.  2,  to  show  which  of  the 
two  directions  of  the  line  is  to  be  taken.  When  forces  repre- 
sented by  lines  lettered  at  their  extremities  are  referred  to, 
the  order  of  the  letters  indicates  the  direction  of  action. 
Thus,  a  force  A  B  means  a  force  acting  from  A  toivards  B. 

56.  The  Parallelogram  of  Velocities  may  be  stated  as  fol- 
lows :  — 

If  two  velocities  be  represented  in  magnitude  and  direction 
by  two  straight  lines  drawn  from  any  point,  the  diagonal  of  the 
parallelogram  constructed  upon  these  two  lines  will  represent 
the  resultant  velocity  in  magnitude  and  direction. 

This  proposition  is  one  of  the  utmost  importance,  inasmuch 
as  the  great  majority  of  mechanical  problems,  whether  of  the 

Sractical  kind  which  require  solution  in  the  operations  of 
ivil  Engineering,  or  of  a  purely  theoretical  nature,  involve 
the  action  of  several  forces  in  different  directions. 

Mr.  Stewart  has  given,  in  Lesson  IV.,  §  24,  a  partial  proof 
for  the  case  of  two  simultaneous  continuously  acting  forces. 


ANSWERS  AND  SOLUTIONS.  129 

The  proof  there  given  is  imperfect,  because  it  is  only  shown 
that  the  body  will  be  at  the  extremity  of  the  diagonal  at  the 
end  of  the  time  considered,  not  that  it  will  constantly  be  on 
this  diagonal  throughout  the  interval.  This  is  a  very  natural 
inference,  it  is  true,  but  it  is  an  inference  which  can  be  de- 
duced legitimately  from  principles  more  general,  if  not  more 
axiomatic  in  their  character. 

In  the  example  of  composition  of  velocities  selected  by  Mr.  - 
Stewart,  the  velocities  are  generated  by  gravity  and  by  mag- 
netic attraction,  —  that  is,  by  continuously  acting  forces,  — 
and  consequently  the  velocities  generated  are  not  uniform  but 
accelerated.  In  other  words,  Mr.  Stewart's  proposition  is  not 
the  Parallelogram  of  Velocities  but  of  Accelerations,  —  a  prop- 
osition equally  true,  but  usually  and  properly  considered  a  logi- 
cal deduction  from  the  more  simple  case  of  uniform  motion. 

Sir  Isaac  Newton,  taking  the  case  of  uniform  velocities,  has 
given  in  his  "  Principia"  the  following  simple  and  convincing 
proof  of  this  famous  proposition.  It  will  be  noticed  that  he 
obtains  the  result  as  a  deduction  from  his  First  and  Second 
Laws  of  Motion. 

Suppose  that  a  body,  in  a  given  time,  by  the  effect  of  a 
single  force  M  impressed  at  A,  would  move  with  a  uniform 

velocity  from  A  to  B  ;  and  A ^ B 

suppose  that  the  body,  in 
the  same  time,  by  the  ef- 
fect of  another  single  force 
N  impressed  at  A  would 
move  with  a  uniform  ve- 
locity from  A  to  C  ;  then 
if  both  forces  act  simul- 
taneously at  A,  the  body 
will  move  uniformly  in 
the  given  time  along  the  YIG.  1. 

diagonal  from  A  to  D. 

For  since  the  force  N  acts  in  the  direction  of  the  line  A  C 
parallel  to  B  D,  this  force,  by  the  Second  Law,  will  not  at  all 
alter  the  velocity  generated  by  the  other  force  M  by  which  the 
body  is  carried  towards  the  line  B  D.  The  body,  therefore,, 
will  arrive  at  the  line  B  D  in  the  same  time  whether  the  force 
^Vbe  impressed  or  not ;  and  therefore  at  the  end  of  that  time 
it  will  be  found  somewhere  in  the  line  B  D.  By  the  same 
6*  I 


130  ELEMENTARY  PHYSICS. 

reasoning,  at  the  end  of  the  same  time  it  will  be  found  some- 
where in  the  line  C  D.  Therefore  it  will  be  found  at  the 
point  D  where  both  lines  meet.  And  it  will  move  in  a 
straight  line  from  A  to  D  by  the  First  Law. 

It  will  be  observed  that  Newton  supposes  that  the  two 
forces  act  instantaneously  ;  that  is,  are  of  the  nature  of  blows, 
as  thJtt  of  a  bat  upon  a  ball.  Such  a  force  communicates  its 
effect  in  a  time  too  small  to  be  taken  account  of,  and  is  in 
this  respect  totally  different  from  a  force  like  gravity  or 
attractions  of  any  kind  which  act  continuously.  Forces  of  the 
former  kind  are  often  called  impulsive  forces,  and  forces 
which  act  during  finite  periods  of  time,  like  gravity,  are 
called  continuous  forces.  Impulsive  forces  tend  to  produce 
uniform  motion,  being  only  prevented  from  so  doing  by  the 
universal  presence  of  retarding  causes,  such  as  friction,  resist- 
ance of  the  air,  &c.  Continuous  forces  tend  to  produce  accel- 
erated motions.  In  Newton's  exposition,  the  uniform  motions 
are  supposed  to  be  produced  in  the  simplest  possible  way,  by 
the  action  of  impulsive  forces,  combined  with  the  absence  of 
retarding  causes.  But  in  nature  no  such  instances  can  occur, 
because  retarding  causes  always  exist.  Bodies  may  move 
with  uniform  velocities,  but  the  uniform  motion  must  arise,  in 
every  case,  from  the  fact  that  several  forces  are  acting  on  the 
body  in  such  a  manner  that  they  neutralize  each  other's  effects 
and  leave  the  body  free  to  obey  the  First  Law  of  Motion.  For 
instance,  a  horse  is  drawing  a  load  along  a  road  at  a  uniform 
rate  ;  here  the  muscular  effort  of  the  horse  is  just  balanced 
by  the  friction  of  the  wheels  on  the  ground,  and  the  resistance 
of  the  air,  which  latter  is,  of  course,  very  trifling. 

In  fact,  the  reference  which  Newton  makes  in  his  proof  to 
force  as  the  cause  of  the  motion  is  unnecessary  ;  and,  inas- 
much as  it  has  been  found  of  great  advantage  in  Mechanics  to 
treat  many  properties  of  "motion,  displacement,  and  deforma- 
tion "  independently  of  force,  mass,  &c.,  under  the  head  of 
Kinematics,  or  the  Geometry  of  Motion,  we  will  present  the 
solution  of  the  Parallelogram  of  Velocities  free  from  any  ref- 
erence to  force. 

Suppose  that  a  body  at  A  (Fig.  1)  moves  with  uniform  ve- 
locity from  A  to  B,  and  that  simultaneously  the  line  A  B  moves 
uniformly,  and  parallel  to  itself,  in  the  direction  A  C  or  B  D. 
Suppose  also  that  in  the  time  required  for  the  body  to  move  from 


ANSWERS  AND  SOLUTIONS. 


131 


A  to  B,  the  line  would  move  from  the  position  A  B  to  the 
position  C  D.  Then  if  the  line  remained  at  rest,  the  body 
would  be  at  B  at  the  end  of  the  time  considered,  but  since  by 
the  motion  of  the  line  the  point  B  takes  the  position  D,  and 
since  both  motions  take  place  independently  of  one  another, 
the  body  will  be  found  at  the  end  of  the  interval  at  the  point 
D.  This  is  true  whatever  be  the  time  considered,  provided 
both  motions  are  uniform.  If,  then,  we  take  one  half  the 
interval  of  time  already  considered,  the  velocities  represented 
by  A  B  and  A  C  will  each  be  reduced  one  half,  and  by  draw- 
ing the  dotted  lines  we  obtain  a  parallelogram  similar  to  the 
first  one  by  Geometry.  Therefore  its  diagonal  is  equal  to 
£  A  D  ;  that  is,  the  time  being  halved,  the  space  passed  over 
by  the  body  is  halved,  the  direction  of  the  body's  motion 
being  unchanged.  The  same  reasoning  may  be  applied  to,  any 
and  every  fractional  part  of  the  time  considered  ;  therefore 
the  body  will  move  uniformly  in  a  straight  line  from  A  to  D. 
58.  Let  the  velocities  be  represented  by  the  lines  0  A,  OB, 
0  C,  and  0  D  in  Fig.  2.  (The  line  0  H  is  not  employed  in 
this  Exercise.) 


FIG.  2. 


132  ELEMENTARY  PHYSICS. 

Find,  by  the  Parallelogram  of  Velocities,  the  resultant  0  E 
of  the  first  two  velocities  0  A  and  0  B.  Then  compound  by 
the  same  principle  this  resultant  with  the  velocity  0  C, 
obtaining  0  F  as  the  resultant  of  the  first  three  velocities. 
Proceeding  in  the  same  way  with  the  remaining  velocity  0  D, 
we  finally  obtain  0  G  as  the  resultant  or  single  velocity 
equivalent  to  all  the  simultaneous  velocities.  This  method  is 
evidently  applicable  to  any  number  of  simultaneous  velocities 
impressed  upon  a  body  at  0. 

It  should  be  observed,  that,  speaking  strictly,  it  is  im- 
possible for  a  body  to  be  at,  or  in,  a  point  0.  Bodies  are 
finite  portions  of  matter  and  have  finite  magnitudes.  And, 
in  point  of  fact,  if  the  several  velocities  above  represented 
were  simultaneously  impressed  on  a  body  in  directions  having 
a  common  point  of  intersection  0  within  the  body,  0  N  would 
not  represent  the  actual  resultant  motion  of  the  body  (unless 
0  wrere  the  centre  of  gravity  of  the  body),  on  account  of  the 
connections  of  the  parts  of  the  body  due  to  cohesion,  arid  the 
consequent  mutual  actions  of  those  parts.  The  determination 
of  the  actual  motion  of  the  body  is  a  problem  of  far  greater 
difficulty  than  the  simple  operation  of  finding  a  resultant  by 
the  Parallelogram  of  Velocities.  But  this  principle  is  a  first 
and  an  essential  step  in  the  chain  of  reasoning  which  leads  to 
the  solution  of  any  case  of  motion  however  complex.  The 
first  thing  to  be  done,  therefore,  is  to  set  forth  the  funda- 
mental principle  in  as  simple  a  form  as  possible.  Most 
writers  on  Mechanics  do  this  by  the  aid  of  the  conception  of 
material  points  or  particles,  which  are  defined  as  bodies  so 
small  that  their  dimensions  may  be  neglected.  Bodies  are 
regarded  as  composed  of  an  indefinitely  large  number  of  parti- 
cles. It  will  be  noticed  that  the  conception  of  a  particle  is 
different  from  that  of  the  molecule  in  Chemistry. 

If,  then,  we  use  language  strictly,  instead  of  speaking  of  a 
"body  at  0,"  we  should  say  a  "particle  at  0,"  or  "a  mate- 
rial point  at  0,"  or,  for  shortness'  sake,  "  a  point  at  0." 

59.  In  Exercise  56  the  object  was  to  find  a  single  velocity 
which  was  equivalent  to  two  simultaneous  velocities ;  this 
is  frequently  called  the  Composition  of  Velocities.  It  is  often 
necessary  for  the  purposes  of  proof  or  illustration  to  perform 
the  reverse  operation,  —  to  substitute  for  a  single  velocity  two 
velocities  in  assigned  directions. 


ANSWERS  AND  SOLUTIONS. 


133 


For  instance,  a  ship  is  sailing  at  the  rate  of  8  miles  per 
hour  in  a  direction  30°  to  the  east  of  south  ;  at  what  rate  is 
she  moving  towards  the  east  and. towards  the  south  respec- 
tively ?  Questions  of  this  kind  fall  under  the  general  problem 
of  the  Resolution  of  Velocities  ;  this  problem  is  solved  geomet- 
rically as  follows  :  — 

Let  0  C  represent  the  velocity  which  we  wish  to  resolve  in 
the  directions  0  X  and  0  Y  ;  from  C  draw  the  lines  C  A  and 
C  B  parallel  to  0  Y  and  OX  Y 
respectively.  Then,  by  the 
Parallelogram  of  Velocities, 
O  C  is  the  resultant  of  ve- 
locities represented  by  0  A 
and  0  B,  so  that  0  A  and 
0  B  are  the  resolved  parts 
required.  Resolution  in  di- 
rections making  a  right 
angle  with  each  other  is  by 
far  the  most  common  and 
useful,  being  attended  with 

this  great  and  obvious  ad-      u  "&  X 

vantage,  that   the   compo-  FIG.   3. 

nents    0  A    and   0  B    are 

wholly  independent  of  each  other,  so  that  each  component 
represents  the  entire  effect  of  the  velocity  0  C,  estimated  in 
its  own  direction.  In  the  question  above  asked,  it  can  be 
easily  found  that  the  ship  is  moving  eastward  at  the  rate  of 
4  miles  an  hour  and  at  the  same  time  moving  southward  at 
the  rate  of  4  V/3  miles  an  hour. 

60.  The  resultant  of  the  velocities  0  A  and  0  B,  in  Fig.  4, 
is  a  velocity  represented  by  the  third  side  0  D  of  the  tri- 
angle 0  A  D.  This  resultant  would  be  neutralized  by  an 
equal  and  opposite  velocity  0  C  impressed  upon  the  partic'e 
at  0,  so  that  0  A,  OB,  and  0  C  form  a  system  of  three 
velocities  such  that,  if  impressed  simultaneously  upon  the 
particle  at  0,  the  particle  would  remain  at  rest.  Now  taking 
the  sides  of  the  triangle  O  A  D  in  order,  0  A  represents  the 
first  velocity,  A  D  represents  the  second,  because  it  is  parallel 
and  equal  to  0  B,  and  DO  represents  the  third  velocity  O  C, 
which  is  by  supposition  equal  to  D  0  and  has  the  same  di- 
rection. 


134 


ELEMENTARY  PHYSIOS. 


FIG.  4. 

61.  This  is  merely  an  extension  of  the  process  of  proof  in 
Exercise  60.     Use  Fig.  2  for  illustration. 

62.  The  term  projection  is  one  of  great  importance  in  cer- 

tain branches  of  study, 
especially  Descriptive  Ge- 
ometry and  Mechanical 
Drawing.  What  we  are 
here  concerned  with  is 
the  projection  of  one  line 
on  another ;  this  may 
be  denned  by  reference  to 
Fig.  5.  Let  A  B  be  a 
line  of  given  length.  To 
_  find  its  projection  on  any 
'  line,  XY,  let  fall  from 
A  and  B  lines  perpendic- 

FlG.  5.  ular  to  X  Y  and  meeting 

X  Y  at  the  points  M  and  N.     Then  M  1ST  is  the  projection  of 
ABonXY. 

Now  it  is  clear  (as  was  stated  in  the  Solution  of  Exercise 
59)  that  the  effective  component  of  a  velocity  in  any  direc- 
tion will  be  found  by  resolving  the  velocity  into  two  com- 
ponents ;  one  in  the  given  direction,  which  will  be  the 
effective  component  sought,  the  other  in  a  direction  perpen- 
dicular to  the  given  direction,  which  component  is  wholly 
independent  of  the  former.  If  we  resolve  A  B  into  these  two 
components  by  the  aid  of  the  Parallelogram  of  Velocities,  we 
find  them  to  be  A  C  and  A  D,  of  which  the  former,  A  C,  is  the 


N 


ANSWERS  AND  SOLUTIONS.  135 

effective  component  in  the  direction  X  Y,  and  is  equal  to  M  N, 
or  the  projection  of  A  B  on  X  Y. 

The  examination  of  the  special  case  referred  to  is  left  to  the 
student. 

67.  A  velocity  may  change,  (1)  in  quantity,  (2)  in  direction. 
We  are  here  concerned  only  with  the  first  kind  of  change,  the 
motion  being  supposed  to  be  rectilinear.  For  curvilinear 
motion,  see  Exercises  124-129.  Acceleration  is  the  rate  of 
change  of  velocity.  Velocity  may  change  in  quantity  either 
by  increasing  or  by  diminishing,  but  the  term  "  acceleration  " 
may  be  extended  to  include  both  cases  by  applying  to  ve- 
locities and  accelerations  the  algebraic  convention  of  positive 
and  negative  signs  to  denote  opposite  directions.  Acceleration 
of  velocity  may  be  either  uniform  or  variable  :  it  is  said  to  be 
uniform  when  the  point  receives  equal  increments  of  velocity 
in  equal  times,  and  is  then  measured  by  the  increase  of  velocity 
per  unit  of  time.  To  illustrate,  take  the  force  of  gravity, 
and  let  the  direction  of  this  force  (towards  the  earth's  centre) 
be  considered  positive.  It  is  known  that  a  body,  let  fall  from 
a  point  above  the  earth's  surface,  acquires  a  velocity  of  32  feet 
per  second  in  the  first  second ;  that  is,  at  the  end  of  the  first 
second  it  is  moving  at  such  a  rate  that  it  would  move  over 
32  feet  during  the  next  second,  if  its  motion  were  uniform. 
But  its  motion  is  not  uniform  ;  gravity  exerts  the  same  effect 
upon  it  during  the  second  as  during  the  first  second,  so  that 
at  the  end  of  the  second  second  its  velocity  is  64  feet  per 
second  ;  and  at  the  end  of  the  third  second  it  would  be  96  feet 
per  second,  and  so  on.  In  this  case  the  acceleration,  or  change 
of  velocity  per  unit  of  time,  is  uniform,  and  is  +32  feet  per 
second  in  each  second.  Which  direction  shall  be  considered 
positive  in  any  case  is  purely  arbitrary,  and  is  usually  selected 
on  grounds  of  convenience.  In  the  example  just  given,  where 
gravity  was  the  only  force  concerned,  and  the  motion  pro- 
duced by  gravity  the  only  motion,  the  simple  and  obvious 
course  was  to  call  the  direction  of  gravity  the  positive  direc- 
tion. But  take  the  familiar  case  of  a  stone  thrown  vertically 
upwards.  This  is  an  example  of  retarded  motion.  But  the 
analysis  of  the  case  shows  that  there  are  two  motions,  simul- 
taneous, independent  of  each  other,  and  in  opposite  direc- 
tions, —  one  a  uniform  motion  vertically  upwards  due  to  the 
muscular  impulse  of  the  hand,  the  other  a  uniformly  acceler- 


136  ELEMENTARY  PHYSICS. 

ated  motion  vertically  downwards  due  to  the  constant  action 
of  the  force  of  gravity.  We  may  take  either  direction  as 
positive  ;  then  the  opposite  direction  must  be  considered  neg- 
ative. It  is  rather  more  convenient,  probably,  to  call  the 
upward  direction  positive  ;  and  it  is  certainly  more  natural 
to  consider  the  direction  of  the  actual  motion  as  positive,  and 
retarded  motion  as  an  instance  of  negative  acceleration.  Let 
this,  then,  be  assumed,  and  suppose  a  stone  thrown  vertically 
upwards  with  a  velocity  of  96  feet  per  second.  It  has  two 
simultaneous  motions,  one  represented  by  the  uniform  velocity 
H-96,  the  other  represented  by  the  uniform  acceleration  — 32. 
It  is  evident  that  at  the  end  of  3  seconds  the  stone  will  have 
a  velocity  of  — 96  downwards,  produced  by  gravity,  which 
will  just  cancel  the  velocity  +96,  that  is,  the  stone  will  come 
to  reSt.  The  space  passed  over  by  the  stone  is  quite  another 
matter,  and  is  investigated  in  Exercises  78  and  80. 

68.  The  average  acceleration  during  any  time  is  the  total 
velocity  gained  during  that  time  divided  by  the  time. 

71.  In  9  seconds  a  body  will  fall  through  1  (9 '8)  81  metres  ; 
in  8  seconds  it  will  fall  through  \  (9*8)  64  metres.  Therefore, 
during  the  ninth  second  it  falls  through  \  (9'8)  (81  —  64) 
metres  =  4*9  X  17  metres  =  83 '3  metres. 

75.  Here,  and  in  78  and  79,  the  notation  is  employed  with 
special  references  to  the  most  important  and  universal  of  all 
forces,  that  of  gravity.  The  initial  velocity  a  is  supposed  to 
be  in  the  direction  of  gravity.  The  letter  g  is  universally 
employed  by  writers  on  physical  science,  as  in  this  Exercise, 
to  denote  the  acceleration  of  gravity,  which  varies  slightly 
with  the  locality,  and  is  only  expressible  approximately  by 
decimals  not  needed  in  most  investigations,  g,  then,  denotes 
the  number  of  units  of  velocity  generated  by  the  force  of 
gravity  in  one  second.  The  force  being  constant,  the  number 
generated  each  second  will  be  the  same,  and  therefore  at  the 
end  of  t  seconds  the  velocity  acquired  by  the  body  will  be  g  t 
units.  This  is  the  entire  effect  of  the  force  of  gravity,  and 
this  effect  is  by  the  Second  Law  of  Motion  the  same  whether 
the  body  be  initially  at  rest  or  moving  in  any  direction  with 
any  velocity.  In  this  case  the  body  is  supposed  to  have 
initially  a  uniform  velocity  of  a  units,  which  velocity  it  wrould 
continue  to  have  forever,  by  First  Law,  were  no  forces  to  act 
upon  it.  It  is  evident,  then,  that  as  the  initial  and  acquired 


ANSWERS  AND  SOLUTIONS.  137 

velocities  are  in  the  same  direction,  the  resultant  velocity  at 
the  end  of  t  seconds  will  be  their  sum,  or  a  -+-  g  t.  This 
formula  includes  the  case  in  which  the  initial  velocity  a,  is 
opposite  to  the  direction  of  g  simply  by  considering  either  a, 
or  g  negative,  as  pointed  out  in  the  Solution  of  Exercise  67. 

78.  We  may  employ  here  the  principle  of  the  Independence 
of  Motions  (or  the  Effects  of  Forces),  stated  by  Mr.  Stewart  at 
the  beginning  of  Lesson  III.,  and  there  called  by  him  the 
Second  Law  of  Motion,  although  the  Law  as  given  by  Newton 
contains,  or  at  least  implies,  much  more.  (See  Solutions  of 
135  and  138.)  Where  several  forces  act  on  a  body,  estimate 
separately  the  effect  of  each  force  in  producing  motion  ;  then 
combine  these  effects  by  addition  or  subtraction,  if  the  forces 
act  along  one  line,  or  by  the  aid  of  the  Parallelogram  of  Veloci- 
ties, if  they  act  along  different  lines.  In  this  case  we  have 
two  motions  along  the  same  line.  The  point  at  the  instant 
when  the  time  t  begins  to  be  reckoned  is  moving  with  a 
uniform  velocity  a  due  to  a  force  (the  impulse  of  the  hand, 
say)  which  has  ceased  to  act.  From  this  instant  the  effect  of 
gravity  is  also  to  be  taken  into  account.  In  virtue  of  the 
uniform  velocity  a  the  point  will  describe  the  space  a  t  in 
the  time  t.  In  virtue  of  the  action  of  gravity,  it  will  de- 
scribe the  space  %gt2  in  the  time  t,  as  shown  by  Mr.  Stew- 
art in  Lesson  III.  Therefore  the  total  space  described  is 
a  t  H-  J  g  t2. 

We  may  prove  this  theorem  without  reference  to  the  Laws 
of  Motion,  as  follows  :  the  average  velocity  of  a  point  during 
any  interval  of  time  is  the  space  described  during  that  time 
divided  by  the  time.  In  this  case,  since  the  acceleration  is 
uniform,  the  average  velocity  is  tjie  arithmetical  mean  between 
the  initial  and  final  velocities.  For  as  the  velocity  increases 
uniformly,  its  value  at  any  time  before  the  middle  of  the  inter- 
val is  as  much  less  than  this  mean  as  its  value  at  the  same 
time  after  the  middle  of  the  interval  is  greater  than  the  mean. 
Here  the  initial  velocity  is  a,  and  the  final  velocity  a  -4-  g  t. 
Therefore  the  average  velocity  =  |  (a  -f-  a  •+•  g  t)  =  a-\-\gt. 
And  space  described  =  average  velocity  X  time  —  a  t  -f-  ^  g  t2. 

80.  Here  the  initial  velocity  is  opposite  to  the  direction  of 
gravity.  Consider  the  direction  of  projection  positive,  and 
that  of  gravity  negative.  Then  the  general  formulae  of  75 
and  78  must  be  written,  — 


138  ELEMENTARY  PHYSICS. 

(1)  v^a  —  gt 

(2)  s  =  at  —  \g& 

The  time  of  ascent  is  evidently  the  value  of  t  in  (1)  when 

v  =•  0,  or  -.     The  height  ascended  is  found  by  substituting 

this  value  of  t  in  (2). 

81.  Put  s  =  0  in  equation  (2)  in  the  preceding  solution, 

and  solve  with  respect  to  t.     We  find  t  =  0,  or  t  =  — .    The 

first  value  corresponds  to  the  instant  of  leaving  the  ground, 
the  second  to  the  instant  of  reaching  it  again.     But  we  have 

already  seen  that  the  time  of  ascent  =  -.     Therefore  time  of 

a  9 

descent  also  =  -. 

82.  If  a  =  velocity  of  projection,  the  time  of  ascent  =  - 
=  time  of  descent,  by  80  and  81.     But  if  a  body  falls  freely 
during  the  time  -,  the  velocity  acquired  =  g  X  -  =  #• 

y  y 

LESSON  IV. 

135.  "We  will  here  give  Newton's  Laws  of  Motion,  both  in 
the  original  Latin  of  Newton,  and  as  translated  by  Thomson 
and  Tait  in  their  Treatise  on  Natural  Philosophy.  Though 
called  by  Newton  Laws  of  Motion,  it  would  be  more  accurate 
to  call  them  Laws  relating  to  the  Connection  of  Force  with 
Motion. 

LEX  I.  Corpus  omne  perseverare  in  statu  suo  quiescendi 
vel  movendi  uniformiter  in  directum,  nisi  quatenus  illud  a 
viribus  impressis  cogitur  statum  suum  mutare. 

LAW  I.  Every  body  continues  in  its  state  of  rest,  or  of  uni- 
form motion  in  a  straight  line,  except  in  so  far  as  it  may  be 
compelled  by  impressed  forces  to  change  that  state. 

LEX  II.  Mutationem  motus  proportionalem  esse  vi  motrici 
impressae  et  fieri  secundum  lineam  rectam  qud  vis  ilia  im- 
primitur. 

LAW  II.  Change  of  motion  is  proportional  to  the  impressed 
force,  and  takes  place  in  tJie  direction  of  the  straight  line  in 
which  the  force  acts. 


ANSWERS  AND   SOLUTIONS.  139 

LEX  III.  Actioni  contrariam  semper  et  aequalem  esse  re- 
actionem :  sive  corporum  duorum  actiones  in  se  mutub  semper 
esse  aequales  et  in  paries  contrarias  dirigi. 

LAW  III.  To  every  action  there  is  always  an  equal  and 
contrary  reaction ;  or,  the  mutual  actions  of  any  two  bodies 
are  ahuays  equal  and  oppositely  directed. 

136.  Change  of  motion  is  determined  in  the, first  place  by 
the  mode  in  which  the  quantity  of  motion  of  a  moving  body  is 
measured.  This  measure  Newton  himself  explicitly  defines  in 
the  second  of  his  Definitions  which  precede  his  Axiomata  sive 
Leges  Motus.  The  quantity  of  motion  or  momentum  of  a  rigid 
body  moving  without  rotation  is,  according  to  Newton,  pro- 
portional to  its  mass  and  velocity  conjointly.  Thus  with  a 
double  mass  and  equal  velocity  the  quantity  of  motion  is 
double  ;  if  the  velocity  be  also  doubled,  it  is  quadruple.  Take, 
as  unit  of  momentum,  that  of  unit  of  mass  moving  with  unit 
of  velocity ;  then  the  momentum  of  m  units  of  mass  moving 
with  v  units  of  velocity  w$ll  be  m  v,  a  result  agreeing  with 
the  definition  of  momentum  given  by  Mr.  Stewart,  §  23.  By 
change  of  motion,  then,  Newton  meant  change  of  momentum, 
and  this  change  may  arise  either  from  change  of  mass  or 
change  of  velocity.  Mass  is  an  element  which  is  not  con- 
sidered in  the  motion  of  material  points ;  in  all  other  cases 
variations  in  mass  simply  produce  proportional  variations  in 
momenta.  What  is  meant  by  change  of  velocity  is  plain 
enough  so  long  as  the  change  takes  place  along  the  line  of 
motion ;  the  change  is  to  be  added  to  the  existing  velocity  if 
it  takes  place  in  the  same  direction,  subtracted  from  this 
velocity  if  it  takes  place  in  the  opposite  direction.  But  sup- 
pose a  velocity  change  in  direction  as  well  as  amount ;  how 
is  this  change  to  be  measured  ?  We  cannot  discuss  this  ques- 
tion here,  but  Fig.  1,  page  129,  will  serve  to  show  the  charac- 
ter of  the  answer.  If  the  velocity  A  B  be  changed  to  A  D, 
the  change  is  represented  in  magnitude  and  direction  by  A  C. 
In  the  example  mentioned  in  the  Solution  of  59,  if  a  ship  be 
sailing  due  east  at  the  rate  of  4  miles  per  hour,  and  its  course 
be  suddenly  changed  so  that  it  begins  to  move  with  a  velocity 
of  8  miles  per  hour  in  a  direction  60°  south  of  east,  the  change 
has  been  4  y  3  miles  per  hour  due  south.  This  represents 
wrhat  has  been  added,  so  to  speak,  to  the  easterly  motion  to 
produce  a  velocity  of  8  miles  per  hour  in  a  direction  60°  south 
of  east. 


140  ELEMENTARY  PHYSICS. 

137.  This  principle  is  given  by  Mr.  Stewart  at  the  beginning 
of  Lesson  111.      It  is  stated  yet  more  generally  by  Thomson 
and  Tait    as  follows  :    "  When  any  forces  whatever  act  on  a 
body,  then,  whether  the  body  be  originally  at  rest,  or  moving 
with   any  velocity  and  in  any  direction,  each  force  produces 
in  the  body  the  exact  change  of  motion  which  it  would  liave 
'produced   if  it   had  acted  singly  on   the  body   originally  at 
rest."     The  student  should  try  to  find  illustrative  examples 
of  the  truth  of  this   principle  additional  to  those  given  by 
the  author. 

138.  The  Second  Law  informs  us  that  a  force  is  propor- 
tional to  the  change  of  motion  which  it  produces,  and  change 
of  motion  has  been  explained  to  be  change  of  momentum  in 
the  Solution  of  136.     Force,  then,  is  to  be  measured  by  the 
change   of  momentum   which   it   produces.     But   as   a  force 
produces   a   continuous  change  of  momentum,  we  can  only 
compare  forces  with  each  other  by  comparing  the  changes  of 
momentum  produced  in  some  one  common  interval  of  time. 
The  simplest  interval,  and  the  one  universally  adopted,  is  the 
unit  of  time,  or  one  second.     Thus  it  appears  that  the  mo- 
mentum generated  in  unit  of  time  is  the  measure  of  a  force, 
and  this  is  equal  to  the  product  of  the  mass  of  the  moving 
body  and  the  acceleration,  because  the  change  of  velocity  per 
unit  of  time  is  by  definition  acceleration. 

139.  The  general  definition  is  a  force  which  acting  for  a 
unit  of  time  upon  a  unit  of  mass  will  generate  a  unit  of  velocity. 

141.  The  statical  unit  of  force  (pound  or  kilogramme)  is 
the  weight  of  unit  of  mass,  or  the  pressure  whicli  unit  of  mass 
exerts  in  consequence  of  the  earth's  attraction  upon  it.  Now 
a  unit  of  mass  falling  freely  acquires  a  velocity  of  g  units  in 
one  second.  Therefore  the  dynamical  measure  of  the  force  of 
gravity  upon  a  unit  of  mass  is  equal  to  1  X  9  or  9  units  of 
force.  In  other  words,  the  same  force  which  is  expressed  in 
statical  measure  by  1  unit  is  expressed  in  dynamical  measure 
by  g  units,  or 

g  dynamical  units  =  1  statical  unit. 

1  statical  unit 
Therefore,  1  dynamical  unit  =  . 

The  derivation  of  the  rules  referred  to  is  left  as  an  exercise 
for  the  student. 


ANSWERS  AND  SOLUTIONS.  141 

142.  The  first  and  fundamental  use  of  the  words  "pound" 
and  "kilogramme"  is  to  denote  units  of -mass  or  quantities  of 
matter  equal  to  that  contained  in  certain  precisely  defined  stand- 
ards preserved  in  the  Public  Archives  with  the  greatest  care. 
The  secondary  use  is  as  units  of  force  or  rather  of  pressure. 
"VVe  are  all  brought  into  daily  contact  with  the  force  of  gravity 
as  producing  pressure  if  not  motion,  and  Engineers,  Architects, 
and  many  other  practical  men  are  constantly  called  upon  to 
measure  and  compare  pressures  produced  by  gravity.     It  was 
very  natural  and  very  convenient,  therefore,  to  adopt  as  a 
unit  of  force  the  pressure  produced  by  the  standard  of  mass, 
or   "standard  weight,"   as  it  is  called,   and  to  employ  this 
unit  in  measuring  all  kinds  of  pressures,   whether  produced 
by  gravity  or  by  the  action  of  other  forces.     Thus,  a  force  of 
20  Ibs.  is  a  force  just  capable  of  sustaining  against  gravity  a 
20-lb.  weight.      "In  all  countries,"  says  Prof.  Maxwell,  "the 
first  measurements  of  force  were  made  in  this  way,  and  a  force 
was  described  as  a  force  of  so  many  pounds'  weight  or  grammes' 
weight.     It  was  only  after  the  measurements  of  forces  made 
by  persons  in  different  parts  of  the  world  had  to  be  compared, 
that  it  was  found  that  the  weight  of  a  pound  or  a  gramme  is 
different  in  different  places,  and  depends  on  the  intensity  of 
gravitation,  or  the  attraction  of  the  earth  ;  so  that  for  purposes 
of  accurate  comparison  all  forces  must  be  reduced  to  dynami- 
cal measure."     When  great  accuracy  is  required  in  expressing 
a  force  in  statical  or  gravitation  measure,  it  is  necessary  to  speci- 
fy the  locality  where  the  observation  is  made  ;  thus,  so  many 
London  pounds  of  force,  so  many  Paris  kilogrammes  of  force. 

143.  The  spring  balance  is  an  instrument  for  measuriiig 
force  ;  for  example,  it  measures  directly  the  force  of  gravity 
upon  a  body  ;  if  the  same  body  be  weighed  by  a  spring  bal- 
ance in  different   latitudes,    it  will   have   different   weights, 
because  the  spring  of  the  balance  will  be  stretched  more  or 
less  by  the  changes  in  the  force  of  gravity.     The    common 
balance  is  an  instrument  for  comparing  quantities  of  matter 
or  masses.     However  much  the  earth's  attraction  upon  a  given 
body  may  vary,  it  will  vary  in  the  same  ratio  upon  the  stand- 
ard weights  employed.     If,  then,  we  put  a  body  in  one  pan 
of  a  balance,  and  equipoise  it  by  placing  standard  weights  in 
the  other  pan,  the  equilibrium  will  be  maintained  in  all  parts 
of  the  earth,  and  everywhere  the  mass  of  the  body  will  appear 
to  be,  as  it  really  is,  invariable. 


142  ELEMENTARY  PHYSICS. 

150.  Apply  the  formula  given  in  Appendix  V.,  at  the  end 
of  Table  II.,  remembering  that  in  this  formula  force  is  sup- 
posed to  be  expressed  in  dynamical  measure,  whereas  in  the 
Exercise  statical  measure  is  used.  This  formula  applies  to 
every  case  of  momentum  generated  by  a  constant  force  like 
gravity,  and  is  true,  since  velocity  =  acceleration  X  time. 
The  use  of  this  formula  for  the  Exercise  shows  at  once  that  the 
mass  of  the  train  is  not  required  in  order  to  find  the  momen- 
tum. 

153.  Since  the  forces  act  on  a  point,  mass  is  eliminated,  and, 
therefore,  the  forces  must  be  as  their  accelerations  respectively. 
Moreover,  they  have  the  same  directions  as  these  accelera- 
tions, the  direction  of  a  force  being  defined  by  the  direction 
of  the  motion  which  it  tends  to  produce.  The  lines  repre- 
senting the  accelerations  may  then  be  considered  to  represent 
the  forces  also,  and  hence  a  single  force  measured  by  the 
resultant  acceleration,  and  in  its  direction,  will  be  the  equiva- 
lent of  any  number  of  simultaneously  acting  forces.  Now  it 
can  be  easily  shown  that  the  propositions  known  as  the  Par- 
allelogram of  Velocities  and  the  Polygon  of  Velocities  hold 
equally  true  of  accelerations.  Indeed,  this  may  be  at  once 
inferred  from  the  fact  that  acceleration  is  merely  a  change  of 
component  velocity  in  a  given  direction  ;  hence  it  is  clear  that 
its  laws  of  composition  and  resolution  must  be  the  same  as 
those  of  velocity.  Therefore  the  laws  of  the  composition  and 
resolution  of  forces  are  the  same. 

LESSON  V. 

Definitions  of  Resultant  and  Balancing  Force.  —  "We  will 
give  here  general  definitions  of  these  important  terms,  which 
are  often  mistaken  for  each  other. 

I.  Resultant.  —  When  any  number  of  forces  act  upon  a 
body,  and  are  not  in  equilibrium,  and  when  there  is  one  force 
capable  of  producing  the  same  effect  as  the  system  of  forces, 
this  one  force  is  called  the  resultant  of  the  system.     - 

II.  Balancing  Force.  —  When  any  number  of  forces  acting 
upon  a  body  are  not  in  equilibrium,  but  are  capable  of  being 
reduced  to  equilibrium  by  the  application  of  a  single  force, 
this  force  is  called  their  balancing  force. 

The  resultant  and  the  balancing  force  of  a  system  evidently 
form  a  pair  of  equal  and  opposite  forces. 


ANSWERS  AND  SOLUTIONS.  143 

167.  The  straight  line  should  be  drawn,  from  the  point  of 
application  of  the  force,  in  the  direction  of  the  force,  and 
containing  as  many  units  of  length  as  there  are  units  of 
force. 

169.  The  principles  here  referred  to  are  generally  regarded 
as  axiomatic  when  the  terms  in  which  they  are  expressed  are 
distinctly  understood.  We  shall  so  regard  them,  and  shall 
now  give  them.  They  should  be  carefully  learned,  on  account 
of  their  subsequent  applications. 

I.  Definition  of  Equal  Forces.     Two  forces  are  said  to  be 
equal  if,  when  ap2)lied  to  the  same  point  in  opposite  directions, 
they  balance  one  another,   or  are  in  equilibrium.     Forces  in 
equilibrium  are  often  called  pressures. 

This  definition  is  only  a  modified  form  of  the  following 
more  general  principle,  which  is  Newton's  Third  Law  of  Mo- 
tion applied  to  forces  in  equilibrium. 

II.  Action   and  reaction  are   always  equal  and  opposite. 
A  table  presses  against  a  book  just  as  much  as  the  book 
against  the  table  ;  if  this  were  not  so,   the  book  and  table 
would  move  either  downwards  or  upwards,  according  as  the 
pressure  of  the  book  or  of  the  table  preponderated. 

III.  If  a  material  point  or  rigid  body  be  acted  on   by  a 
system,  of  forces,  then  the  additional  application  of  a  system  of 
forces  in  equilibrium  will  have  no  effect. 

IV.  A  force  may  be  transmitted  to  any  point  in  the  line  of 
its  action,  without  altering  its  effect  on  a  rigid  body. 

V.  The  tension  of  a  perfectly  flexible,  cord  in  contact  only 
with  perfectly  smooth    surfaces  is  the  same    throughout  its 
length. 

174.    See  153,  and  Solution. 

188.  Two  parallel  forces  are  said  to  be  like  when  they  act 
in  the  same  direction,  unlike  when  they  act  in  opposite  direc- 
tions. 

189.  Let  P  and  Q  be  the  forces  acting  at  A  and  B  respec- 
tively, A  and  B  being  regarded  as  rigidly  connected. 

The  effect  of  the  forces  will  not  be  altered  if  we  apply  two 
equal  forces,  S,  S,  at  A  and  B  acting  along  A  B  in  opposite 
directions  (Axiom  III.).  Compound  them  with  P  and  Q 
respectively,  and  we  obtain  for  the  resultants  X  and  Y.  Pro- 
duce the  lines  of  action  of  these  resultants  till  they  meet  at  D, 
and  draw  D  C  parallel  to  the  lines  A  P  and  B  Q,  meeting 


144 


ELEMENTARY  PHYSICS. 


A  B  at  C.  Transfer  the  resultants  X  and  F  to  D  (Axiom  IV. ), 
resolve  them  along  D  C  and  a  straight  line  through  C  parallel 
to  A  B  ;  each  of  the  latter  components  will  be  equal  to  S,  and 
they  will  act  in  opposite  directions,  and  will  balance  each 
other ;  the  sum  of  the  former  components  will  be  P  -h  Q. 

R' 


\ 


FIG.  6. 


Hence  the  resultant  of  the  forces  P  and  Q  is  P  -f-  Q,  and  acts 
along  a  line  D  C  parallel  to  the  lines  of  action  of  P  and  Q  in 
the  same  direction  ;  so  that  it  may  be  supposed  to  act  at  C. 
A  force  represented  by  C  B/  equal  and  opposite  to  C  R  is  the 
balancing  force. 


ANSWERS  AND  SOLUTIONS. 


145 


To  find  the  position  of  C.  The  triangles  A  P  X  and  D  0  A 
are  similar,  being  equiangular  with  respect  to  each  other,  and 
the  line  A  S  =  the  line  P  X. 

TJTTvp  *?     '          "R  P  P  B    C 

Therefore  - ,  =  ^.     Similarly  -  =  ^.     Therefore  -  =  -^ 

190.  This  theorem  may  be  demonstrated  in  precisely  the 
same  way  as  the  preceding,  only  the  changed  conditions  re- 
quire a  separate  figure.  The  student  should  draw  the  figure 
and  go  through  with  the  demonstration. 

Or,  we  may  reason  thus,  using  Fig.  6  and  its  conditions : 
The  forces  P,  Q,  and  R'  =  P  -f-  Q.  form  a  system  of  three 
parallel  forces  in  equilibrium.  Of  these  P  and  R*  form  a  pair 
of  unlike  parallel  forces  held  in  equilibrium  by  Q,  which  is 
therefore  by  definition  their  balancing  force  (see  beginning 
of  solutions  of  exercises  upon  this  Lesson)  ;  and  a  force  equal 
to  Q,  and  acting  in  the  opposite  direction  at  the  point  B,  is 
their  resultant.  We  have  proved  that  P  :  Q  =  B  C  :  A  C. 
By  theory  of  proportions,  P  -+-  Q  :  P  =  (B  C  -h  A  C)  or  A  B  : 
B  C,  which  proves  the  proposition. 

193.  For  it  is  evident  that  the  two  preceding  proofs  hold 
equally  good,  whatever  be  the  angle  between  the  lines  of  action 
of  the  forces  and  the  line  drawn  across  them.     In  Fig.  6  this 
angle  is  a  right  angle,  but  this  is  not  necessary. 

194.  Let  P,  Q,  S,  T,  be  the  parallel  forces  acting  at  the  points 
A,  B,  C,  D,  respectively.    Join  A  B,  and  find  a  point  L  on  A  B 
such  that  P  :  Q  =  B  L  : 

A  L.     Then   L  is  the  n 

point  of  application  of 
the  resultant  of  P  and 
Q,  which  resultant  is 
equal  to  P  •+•  Q.  Join 
L  C,  and  in  the  same 
way  find  M  the  point 
of  application  of  P  •+• 
Q  acting  at  L  and  S 
acting  at  C  ;  this  re- 
sultant will  be  P  -+•  Q 
•+•  S.  Proceeding  in 
the  same  way  with  T  FIG.  7. 

we  find  N  the  point  of 

application  of  P  -h  Q  -H  S  4-  T  =  R,  the  resultant  of  the  four 
7*  J 


146 


ELEMENTARY  PHYSICS. 


parallel  forces.     If  any  of  the  forces  act  in  the  opposite  direc- 
tion, the  process  of  190  must  be  used. 

195  If  the  parallel  forces  in  Fig.  7  were  all  turned  through 
the  same  angular  distance  about  their  points  of  application 
in  such  a  manner  that  their  lines  of  action  remained  parallel, 
it  can  be  easily  seen  that  the  point  of  application  N  of  their 
resultant  would  remain  unchanged,  and  the  resultant  would 
be  turned  about  N  through  the  same  angular  distance.  This 
is  true  whatever  be  the  number  of  the  forces,  and  this  remark- 
able constancy  in  position  of  the  point  of  application  of  the 
resultant  of  any  number  of  parallel  forces  has  caused  it  to  be 
called  the  centre  of  the  system  of  parallel  forces. 

198.  A  couple  is  a  system  of  two  equal,  unlike,  parallel 
forces.  Its  arm  is  the  distance 
between  the  lines  of  action  of 
the  two  forces.  Its  moment  is 
the  product  of  either  force  into 
the  arm,  and  is  usually  con- 
sidered positive  or  negative  ac- 
cording as  it  tends  to  produce 
rotation  opposite  to,  or  the  same 
as,  that  of  the  hands  of  a  watch. 
Thus  in  Fig.  8,  P,  P,  are  the 
forces,  A  B  the  arm,  P  X  A  B 
the  moment  which  in  this  case 
is  negative. 

199.  The  moment  of  a  force 
with  respect  to  a  point  is  equal 
to  the  product  of  the  force  into 
the  distance  from  the  point  to  the  line  of  action  of  the  force. 
The  distinction  between  positive  and  negative  moments  is  the 
same  as  that  between  positive  and  negative  couples.  For 
example  (Fig.  9),  P  is  the 
force  agting  in  the  direc- 
tion  0  P.  Then  its  mo- 
ment  with  respect  to  the 
point  A  is  P  X  A  B  ;  its 
moment  with  respect  to  A' 
is  P  X  A'  B'  ;  its  moment 
with  respect  to  A"  (or  any 


PNl 


Fia  8. 


f 


point  on  its  line  of  action) 
is  P  X  Q  ==  0. 


FIG. 


ANSWERS  AND  SOLUTIONS. 


147 


The  doctrine  of  moments  holds  a  position  of  the  first  im- 
portance in  the  mechanics  of  rigid  bodies,  for  the  reason  that 
a  moment  is  the  appropriate  measure  of  the  tendency  of  a  force 
to  produce  rotation. 

200.  "The  moment  of  a  force  round  any  axis  is  the  mo- 
ment of  its  component  in  any  plane  perpendicular  to  the  axis, 
round  the  point  in  which  the  plane  is  cut  by  the  axis.     Here 
we  imagine  the  force  resolved  into  two  components,  one  par- 
allel to  the  axis,  which  is  ineffective  so  far  as  rotation  round 
the  axis  is  concerned  ;  the  other  perpendicular  to  the  axis, 
that  is  to  say,  having  its  line  in  any  plane  perpendicular  to 
the  axis.     This  latter  com- 
ponent  may   be   called    the 

effective  component  of  the 
force  with  respect  to  rotation 
round  the  axis.  And  its  mo- 
ment round  the  axis  may  be 
defined  as  its  moment  round 
the  nearest  point  of  the  axis, 
which  is  equivalent  to  the 
preceding  definition."  (Thom- 
son and  Tait.) 

201.  Let  (Fig.   10)  P,  Q, 
and  R   (=  P  +  Q)   be  the 
forces   acting   at  the  points 
A,  B,  C,  respectively.   Choose 
any  point  0  in  the  plane  of 
the  forces  ;  join  0  B,  the  line 
0  B  cutting  the  lines  of  P 
and  R  at  A'  and  C'.     Then 
we  have 


FIG.  10. 


x  0  C'  =  (P  +  Q)  x  0  C', 

=  P  x  (0  A'  +  A'C')  +  Q  x  OC', 


since,  by  193,  P  x  A'  C'  =  Q  x  B  C'. 

Therefore,  P  x  0  A'  -t-  $  x  OB  —  P  x  0  C'  =  0, 

the  first  side  of  which  equation  is  the  algebraic  sum  of  the 
moments  of  the  forces. 


148 


ELEMENTARY  PHYSICS. 


202.  Let  Pt,  P2,  P3,  &c.,  be  the  forces.  Draw  any  line 
across  their  lines  of  action,  and  in  it  choose  a  point  of  refer- 
ence, 0.  Let  0  Ax  =  av  0  A2  =  &2,  0  A3  =  a3,  &c. 


FIG.  11. 

First  find  by  means  of  193  the  resultant  of  P,  and  P2 ;  if 
we  denote  it  by  IV,  we  have  R1  =  Pl  +  P2.  Divide  At  A2 
into  parts  inversely  as  the  forces,  so  that 

Pl  x  Aj  Xj  =2  P  x  A2  Xr 
If  we  denote  OX,  by  as'  we  have 

Pi  x  (x'  —  a,)  =  P2  x  (rt,  —  a'), 
or,  (Pt  +  P2)  x'  =Plal  +  P2  «2, 

that  is,  R'  x'  =  Plal  +  P2  av 

Similarly  we  should  find  the  resultant  of  R*  and  Ps  to  be 
Rii  =  pt  +  P2  +  ps,  and  that 

P,'/  x"  =Rfx'  +  P3  ^8  =  Pt  a,  +  P2  «,  +  P,  r^3. 

Hence,  finally  we  have  the  two  equations  given  in  the  state- 
ment of  the  Exercise. 


ANSWERS  AND  SOLUTIONS.  149 

Negative  forces  or  negative  values  of  any  of  the  quantities 
at,  a»,  &c.,  are  included  in  this  method,  provided  the  gen- 
eralized rules  of  multiplication  and  division  in  algebra  are 
followed. 

203.  The  necessary  and  sufficient  conditions  for  the  equi- 
librium of  any  number  of  parallel  forces  in  one  plane  are  the 
following  :— • 

(1)  algebraic  sum  of  the  forces  =  0. 

(2)  algebraic  sum  of  the  moments  of  the  forces  round  any 

point  in  the  plane  =  0. 

If  equation  (1)  does  not  hold,  but  equation  (2)  does,  the 
forces  have  a  single  resultant  whose  line  of  action  passes 
through  the  origin  of  co-ordinates. 

If  equation  (1)  does  hold,  but  equation  (2)  does  not  hold, 
the  system  reduces  to  a  couple  ;  this  is  indicated  by  the  fact 
that  x  is  equal  to  infinity  (since  R  x  is  a  finite  quantity,  arid 
JR  by  hypothesis  is  zero)  ;  in  other  words,  the  point  of  appli- 
action  of  the  resultant  is  at  an  infinite  distance,  which  is  the 
case  with  a  couple. 

204.  In  the  first  kind  of  Levers,  the  fulcrum  is  between  P 
and  W ;  in  the  second  kind,   W  is  between  P  and  the  ful- 
crum ;  in  the  third  kind,  P  is  between  W  and  the  fulcrum. 

207.  The  mechanical  advantage  of  any  mechanical  contriv- 
ance or  combination  whatsoever  is  expressed  by  the  fraction 

W 

— ,  P  being  the  applied  force  or  power,  W  the  weight  which 

is  raised  or  pressure  which  is  exerted,  and  friction,  &c.,  being 
neglected.  In  the  case  of  the  lever  this  ratio  is  often  called 
by  workmen  the  leverage.  There  exists  a  popular  impres- 
sion, that  a  machine  can  generate  or  create  force.  This  im- 
pression arises  from  the  well-known  fact  that  by  the  aid  of 
a  machine  an  enormous  weight  can  be  raised,  or  resistance 
overcome,  by  the  application  of  a  very  small  power.  But  on 
examination  it  will  be  found  that  in  the  exact  proportion  in 
which  a  machine  diminishes  the  power  required  to  raise  a 
given  weight,  it  increases  the  distance  through  which  this 
power  must  act  in  order  to  raise  the  weight  through  a  given 
height.  Take  a  lever  the  arms  of  which  are  as  10  to  1  ;  we 
have  seen  that  a  pound  weight  hung  at  the  extremity  of  the 
longer  arm  will  balance  10  pounds  at  the  end  of  the  shorter 


150  ELEMENTARY  PHYSICS. 

arm  ;  the  slightest  addition  to  the  pound  weight,  or  rather, 
friction  being  neglected,  the  slightest  exterior  impulse,  as  a 
touch  with  the  finger,  will  cause  the  pound  weight  to  de- 
scend and  raise  the  10  pounds.  But  a  little  geometrical 
reflection  will  make  it  obvious  that,  in  order  that  the  10  pounds 
may  rise  1  inch,  the  pound  weight  must  descend  10  inches.  A 
more  complex  yet  familiar  instance  is  furnished  by  the  mech- 
anism of  a  watch.  Here  the  power  communicated  by  the 
main-spring  is  applied  to  a  train  of  wheels,  and  produces  a 
much  more  rapid  movement  in  the  balance-wheel.  But  the 
slightest  touch  of  the  finger  will  check  the  balance-wheel, 
while  the  winding  up  of  the  main-spring  requires  a  far  greater 
force.  And  on  examination  it  would  be  found  that,  making 
allowance  for  friction,  the  force  with  which  the  balance-wheel 
moved  was  precisely  as  many  times  less  than  that  required  to 
move  the  main-spring  as  its  motion  was  more  rapid.  In  fact, 
one  grand  and  simple  law  is  rigorously  fulfilled  by  every  ma- 
chine and  mechanical  combination,  however  complex.  This 
law,  if  we  neglect  friction,  may  be  stated  as  follows  :  — 

In  every  machine  the  product  of  the  power  and  the  distance 
through  which  the  power  moves  in  its  own  direction  is  equal  to 
the  product  of  the  weight  (or  resistance  overcome]  and  the  dis- 
tance through  which  the  weight  moves  in  its  own  direction. 

This  is  a  modified  form  of  the  celebrated  principle  of  Vir- 
tual Velocities  ;  it  may  now  be  more  appropriately  termed  the 
Principle  of  Work.  In  fact,  adopting  the  idea  and  measure 
of  work,  explained  by  Mr.  Stewart  in  Chapter  III.,  this  prin- 
ciple is  stated  simply  by  saying  that,  friction  being  neglected, 
the  work  transmitted  by  a  machine  is  unaltered  in  amount ;  or 
more  generally,  friction  being  included,  in  every  machine,  the 
parts  of  which  are  moving  uniformly,  the  work  done  by  the 
power  —  the  work  done  against  the  resistance  +  the  work  done 
against  friction. 

Thus  it  appears  that  a  machine  may  increase  or  diminish 
the  magnitude  of  the  power  in  any  given  ratio,  but  that  it 
cannot  increase  the  product  of  the  force  acting  at  any  part 
into  the  space  through  which  this  force  moves,  or,  in  other 
words,  it  cannot  increase  the  work  transmitted  by  the  machine. 
In  point  of  fact,  this  product  is  diminished  by  the  effect  of 
friction  as  the  force  is  transmitted  through  the  machine-  The 
use  of  a  machine  is  to  apply  more  advantageously  the  force 


ANSWERS  AND  SOLUTIONS. 


151 


applied  to  it,  to  transmit  force,  and  to  change  in  a  desired  de- 
gree the  direction  and  velocity  of  motion. 

211.  Let  a  and  b  denote  the  unequal  arms,  and  x  the  true 
weight  of  the  body.  Put  the  body  on  the  pan  attached  to  the 
arm  «,  and  balance  it  by  weights  in  the  other  pan  ;  call 
these  weights  W.  Next  reverse  the  operation  by  putting  the 
body  on  the  pan  attached  to  the  arm  6,  and  balancing  it  by 
weights  W  on  the  pan  attached  to  the  arm  a.  Then  W  and 
Wf  are  the  two  false  weights.  Now  in  the  first  operation  we 
have,  by  205,  — 

ax  =  Wb, 

and  in  the  second  operation  similarly  we  have 
bx  =  W>  a. 


Hence  a  b  x2  =  IP 

or  x2  =  W  W. 

And  x  =  \/  W  W. 

215.  The  Wheel  and  Axle  is  essentially  a  Lever  of  the  first 
kind,  the  fulcrum  being  the  centre  of  the  axle,  and  the  power 
and  weight  being  applied  at  the  extremities  of  radii  of  the 


FIG.  12. 

wheel  and  the  axle  respectively.     This  is  easily  seen  by  an 
inspection  of  Fig.  12,  which  represents  a  common  Windlass. 


152 


ELEMENTARY  PHYSICS. 


Here  in  place  of  a  wheel  we  have  what  is  equivalent,  four 
arms  or  spokes,  which  are  turned  by  hand.  The  proportion 
given  in  the  statement  of  the  Exercise  can  be  readily  obtained 
by  the  student. 

The  Wheel  and  Axle  possesses  an  important  advantage  over 
the  simple  Lever,  in  the  fact  that  it  allows  us  to  raise  a  weight 
through  any  given  height.  With  a  simple  Lever  a  weight  can 
be  raised  only  a  small  distance  before  it  becomes  necessary  to 
place  the  Lever  in  a  new  position,  and  to  support  the  weight 
by  some  other  force  while  this  change  is  being  made.  The 
Wheel  and  Axle  is  a  practical  arrangement  for  continuing  the 
action  of  a  Lever  as  long  as  may  be  required,  the  weight  rising 
all  the  time. 

217.  One  simple  principle  will  explain  the  mechanical  ad- 
vantage of  any  system  of  Pulleys,  however  complex,  namely, 
the  principle  of  the  tension  of  a  cord,  given  in  the  solution  of 
169.    The  single  movable  Pulley  is  shown  in  Fig.  13.    There 

is  only  one  cord,  which,  by  the  principle 

A"?        L  /CB    referred  to,  has  the  same  tension  through- 

out (friction,  &c.,  being  neglected).  The 
weight,  W,  is  equally  supported  by  the 
parts  of  this  cord  in  contact  with  the 
Pulley  at  A  and  B.  Therefore  W  = 
2  X  tension  of  the  cord.  But  it  is 
evident  that  P  =  tension  of  the  cord. 
Therefore,  W  =  2  P. 

The  single  movable  Pulley  may  also 
be  regarded  as  a  Lever  with  equal  arms, 
and  the  fulcrum  at  the  centre  of  the  Pul- 
FIG.  13.  ley-     The  forces  acting  on   equal  arms 

must  be  equal,  and  therefore  the  press- 
ure on  the  fulcrum  is  twice  the  tension  of  the  cord,  or  2  P. 
But  W  causes  this  pressure  ;  therefore,  W  —  2  P. 

218.  Fig.  14  represents  the  three  systems  of  Pulleys  referred 
to  in  Exercises  218-220.     Of  these  systems  (1)  is  the  sim- 
plest, and  also  the  most  convenient  in  use.     We  will  solve 
case  (2).     In  this  system  there  are  in  general  n  (here  5)  mov- 
able Pulleys,  each  hanging  by  a  separate  cord  as  shown  in  the 
figure.     The  tension  of  the  cord  which  passes  under  the  highest 
movable  Pulley  A  is  by  the  principle  of  tension  equal  to  the 
power  P.     By  the  same  reasoning  which  was  used  in  the  case 


ANSWERS  AND  SOLUTIONS. 


153 


of  the  single  movable  Pulley,  the  tension  of  the  cord  pass- 
ing under  the  next  lower  Pulley  is  2  P,  of  the  cord  passing 
under  the  third  Pulley  is  4  P  =  22  P,  of  the  next  cord  is  23  P, 
of  the  cord  passing  under  the  lowest  Pulley  is  2*  P.  The  ten- 
sion of  this  last  cord  is  by  the  same  reasoning  equal  to  ^  W. 
Therefore,  W  =  25  P.  And,  in  general,  W  =  2»  P. 


223.  The  Inclined  Plane  is  a  plane  inclined  at  any  angle  to 
the  horizon.  The  principle  of  the  Inclined  Plane  consists  in 
tnis,  that  a  weight  W  can  be  supported  on  the  Inclined  Plane 
by  a  power  P  which  is  less  than  W.  It  is  a  direct  example 
of  the  Parallelogram  of  Forces. 

Conceive  that  the  Plane  is  perfectly  smooth,  and  likewise 
the  body  which  is  resting  on  the  Plane.  This  body  has  its 
centre  of  gravity  at  G,  so  that,  as  will  be  explained  later,  its 
weight  W  may  be  regarded  as  entirely  concentrated  in  a 
heavy  point  at  G.  Three  forces  are  in  action  when  the  body 
is  in  equilibrium,  the  weight  of  the  body  W  acting  at  G  ver- 
tically downwards,  the  power  P  applied  to  keep  the  body  at 
rest  on  the  Plane  and  acting  from  G  up  the  Plane,  and  the 
reaction  R  of  the  Plane  arising  from,  the  pressure  of  the  body 


154 


ELEMENTARY  PHYSICS. 


on  the  Plane  and  acting  in  a  direction  perpendicular  to  the 
Plane.     Substitute  for  W  its  two  components  G  M  and  G  N, 


\ 


W 


FIG.  15. 

parallel  and  perpendicular  respectively  to  the  Plane.  These 
components  are  opposite  to  P  and  R  respectively  ;  hence  they 
must  be  equal  to  them  each  to  each,  or,  G  M  =  P,  and 
G  N  =  R.  The  triangles  ABC  and  G  W  N  are  similar ; 
whence  it  follows  that 


and  that 


P^  _  BC  =  height 
W~  AB       length ' 


R 


AC 
AB 


length  * 


226.  Let  the  student  represent  the  conditions  of  this  ques- 
tion by  a  diagram  similar  to  that  given  in  the  solution  of  223. 
He  will  see  that  the  lines  representing  the  three  forces  P,   W9 
and  R  form  a  right  triangle.     This  triangle  may  be  shown  to 
be  similar  to  the  large  triangle  the  hypothenuse  of  which  is 
the  Plane.     Hence  the  required  proportion  is  easily  obtained. 

227.  The  Screw  is  a  movable  inclined  plane  wrapped  round 
a  right  cylinder.    The  relation  between  the  Inclined  Plane  and 
the  Screw  may  be  understood  by  reference  to  Fig.  16. 


ANSWERS  AND  SOLUTIONS. 


155 


Suppose  that  we  unroll  paper  previously  rolled  round  the 
right  cylinder,  A  B  K  L,  by  causing  the  cylinder  to  revolve 
on  its  axis  through  exactly  one  revolution  ;  we  shall  obtain  a 


rectangle,  B  A  D  F.  Divide  this  rectangle  into  a  certain 
number  of  equal  rectangles  by  drawing  lines  parallel  to  the 
base,  A  D  ;  draw  diagonals  to  these  rectangles  as  shown  in  the 
figure  ;  and  lastly,  roll  the  entire  rectangle  again  upon  the 
cylinder.  The  diagonals  will  form  a  continuous  curve  called 
a  helix,  composed  of  spirals,  one  above  another,  equal  in  num- 
ber to  the  number  of  partial  rectangles  (in  this  case  two). 

In  rolling  up  the  rectangle  the  line  mp  for  example  falls 
into  the  position  M  P,  the  point  D  falls,  after  one  revolution, 
upon  A,  the  point  C  falls  upon  E,  and  F  upon  B.  If,  now, 
we  take  a  cylinder  (of  wood  or  iron)  having  on  its  surface,  in 
place  of  a  helix,  traced  in  pencil  or  ink,  a  projecting  thread 
making  at  all  points  the  same  given  angle  (called  the  pitch  of 
the  Screw)  with  the  horizon,  we  have  a  Screw  ready  for  use. 

Thus,  the  Inclined  Plane  A  C  forms  one  revolution  of  the 
helix,  its  base  is  equal  to  the  circumference  of  the  cylinder, 
and  its  height  to  the  distance  between  two  threads,  a  quantity 
of  special  importance  in  the  theory  of  the  Screw. 

228.  In  order  to  adapt  the  Screw  to  use,  a  nut  must  be 
employed;  this  is  a  block  pierced  with  an  equal  cylindrical 
aperture,  upon  the  inner  surface  of  which  is  cut  a  groove  the 


156  ELEMENTARY  PHYSICS. 

exact  counterpart  of  the  thread  of  the  Screw.  It  is  evident 
that  the  Screw  can  only  be  made  to  move  in  the  nut  by  revolv- 
ing about  its  axis.  Suppose  the  Screw,  with  its  axis  vertical, 
to  be  contained  in  the  nut  without  friction,  and  let  a  weight, 
W,  be  placed  upon  it ;  then  the  pressure  due  to  this  weight 
will  be  transmitted  to  every  point  of  the  thread  of  the  Screw. 
Suppose  that  there  are  n  points  on  the  thread,  each  point  in 
contact  with  a  corresponding  point  of  the  groove  in  the  nut ; 

then  each  point  is  acted  upon  by  a  vertical  force  equal  to  — . 

Now  it  is  clear  that  under  these  circumstances  the  Screw  would 
descend  by  revolving  on  its  axis  (that  is,  the  various  points 
of  the  thread  would  slide  down  the  groove  with  a  spiral  mo- 
tion), unless  prevented  by  some  force ;  call  this  force  Pv  and 
let  it  act  at  some  point  on  the  circumference  of  the  cylinder 
along  a  horizontal  line  tangent  to  the  circumference.  This 
force  P.  will  likewise  be  distributed  along  the  thread,  each 

p 

point  being  acted  upon  by  a  horizontal  force  equal  to  — .  Con- 
sider now  the  equilibrium  of  any  point  A  of  the  thread.  It  is 

P      W 
kept  at  rest  bv  three  forces,  —  .  — ,  and  the  reaction  of  the 

n      n 
thread,  which  is  equal  and  opposite  to  the  normal  pressure  on 

P  W 

the  thread  arising  from  the  combined  action  of  —  and  — , 

n  n 

and  which  need  not  be  further  considered  in  this  investiga- 
tion. Bearing  in  mind  the  analogy  between  the  Screw  and 
the  Inclined  Plane,  it  is  evident  that  this  case  is  similar  to  that 
treated  in  226,  and  considered  as  an  instance  of  the  Inclined 

P      W 
Plane,  we  have,  —  :  —  =  height  of  plane  :  base  of  plane. 

n      n 

Simplifying  the  first  ratio,  and  substituting  for  the  terms  of 
the  second  ratio  their  equivalents  in  terms  of  the  Screw,  we 
obtain  P,  :  W  =  distance  between  two  threads  :  circumfer- 
ence of  the  cylinder. 

Practically,  the  Screw  is  never  used  as  a  simple  machine, 
the  power  being  applied  by  means  of  a  Lever  passing  either 
through  the  head  of  the  Screw  or  through  the  nut.  The  Screw 
acts,  therefore,  with  the  combined  power  of  the  Lever  and  the 
Inclined  Plane  ;  and  in  investigating  the  effect  we  must  take 


ANSWERS  AND  SOLUTIONS. 


157 


into  account  both  these  Mechanical  Powers.  The  proportion 
just  given  represents  the  effect  of  the  Inclined  Plane.  To  com- 
bine this  with  the  effect  of  the  Lever  is  left  as  an  exercise  for 
the  student.  The  result  is  stated  on  page  91. 


FIG.  17. 

To  produce  pressure  with  the  Screw  either  the  Screw  or  the 
nut  must  be  fixed  ;  whichever  is  free  is  then  turned,  and  made 
to  press  against  the  resistance.  It  is  evident  that  one  revo- 
lution causes  the  Screw  (or  the  nut)  to  advance  by  an  amount 
equal  to  the  distance  between  two  threads.  In  the  common 
Screw-press,  represented  in  Fig.  17,  the  nut  is  fixed  and  the 
Screw  movable.  A  is  the  Screw,  enlarged  below  at  C,  where 
it  is  pierced  with  two  holes  at  right  angles  for  applying  the 
Lever.  B  is  the  nut  firmly  fixed  in  the  upper  part  of  the 
press.  By  turning  the  Screw  in, the  nut  it  descends  and 
pushes  before  it  the  plate  D,  which  works  in  guides  so  that  it 
can  have  only  a  vertical  motion.  The  body  is  compressed  by 
placing  it  on  the  fixed  platform  E,  and  lowering  the  Screw 
and  plate  D. 


158  ELEMENTARY  PHYSICS. 

In  deducing  the  law  of  equilibrium  of  the  Screw  we  have 
neglected  the  effect  of  friction  for  the  sake  of  simplicity.  But 
the  amount  of  friction  in  every  Screw  is  very  great ;  in  fact, 
the  Screw  owes  its  utility  to  friction  ;  for  if  there  were  no  fric- 
tion the  Screw  would  overhaul,  that  is,  turn  backwards  the  in- 
stant the  power  was  removed.  To  prevent  this  the  friction  must 
be  sufficient  to  consume  more  than  half  the  power  applied. 

Note.  —  All  the  Exercises  upon  the  Mechanical  Powers  can 
be  very  readily  solved  by  means  of  the  principle  of  Work  stated 
in  the  solution  of  207.  The  student  should  first  work  out 
these  Exercises  by  the  methods  already  given  or  indicated, 
and  then  solve  them  by  the  aid  of  the  principle  of  Work.  The 
greater  simplicity  of  this  latter  method  is  strikingly  shown  in 
the  case  of  the  Screw  ;  indeed,  it  is  obvious  that,  friction  being 
neglected,  this  beautiful  principle  may  be  employed  with 
equal  ease  in  any  machine,  whatever  its  nature  and  however 
complex  its  mechanism. 


LESSON  VI. 

271.  For  the  Third  Law  of  Motion,  see  Solution  of  135. 
Newton  gives  three  illustrations. 

1.  "  If  you  press  a  stone  with  your  finger,  the  finger  is  also 
pressed  by  the  stone."     This   is  an  illustration   of  forces  in 
equilibrium  (see  Solution  of  169). 

2.  "  If  a  horse  draws  a  stone  tied  to  a  rope,  the  horse  (if  I 
may  so  say)  will  be  equally  drawn  back  towards  the  stone  ; 
for  the  distended  rope,  by  the  same  endeavor  to  relax  or  un- 
bend itself,  will  draw  the  horse  as  much  towards  the  stone  as 
it  does  the  stone  towards  the  horse,  and  will  obstruct  the  pro- 
gress of  the  one  as  much  as  it  advances  that  of  the  other." 

3.  "  If  a  body  impinge   upon   another,   and   by  its   force 
change  the  motion  of  the  other,  that  body  also  (because  of  the 
equality  of  the  mutual  pressure)  will  undergo  an  equal  change, 
in  its  own  motion,  towards  the  contrary  part.     The  changes 
made  by  these  actions  are  equal,  not  in  the  velocities,  but  in 
the  quantities  of  motion  (momenta)  of  the  bodies." 

272.  "Every  force  acts  between  two   bodies,  or  parts  of 
bodies.     If  we  are  considering  a  particular  body  or  system  of 
bodies,  then  those  forces  which  act  between  bodies  belonging 


ANSWERS  AND  SOLUTIONS.  159 

to  this  system  and  bodies  not  belonging  to  the  system  are  called 
External  Forces,  and  those  which  act  between  the  different  parts 
of  the  system  itself  are  called  Internal  Forces."  (Maxwell.) 

The  same  force  may  play  the  part  of  an  interior  or  exterior 
force  according  to  the  point  of  view  adopted.  Take,  for  in- 
stance, the  motion  of  a  body  which  falls  to  the  earth  ;  the 
attraction  of  one  of  the  particles  of  this  body  for  a  particle 
of  the  earth  is  an  exterior  force  ;  if,  on  the  other  hand,  we 
consider  the  motion  of  the  material  system  formed  of  the  body 
and  the  earth,  this  same  attraction  becomes  an  interior  force. 
The  student  should  find  other  illustrations. 

279.  For  this  distinction,    see  the  solution  of  56.      The 
student  should  give  illustrations.     The  duration  of  the  action 
of  an  impulsive  force  is  too  brief  to  be  measured,  so  that  we 
cannot  measure  the  force  by  the  momentum  generated  in  one 
second.     An  impulsive  force  is  measured  by  the  total  momen- 
tum -which  it  generates. 

280.  A  body  is  said  to  be  perfectly  elastic  if,  when  it  im- 
pinges perpendicularly  on  a  fixed  plane,  it  will  recoil  back  from 
the  plane  with  equal  velocity,  or  when  the  velocity  of  recoil  is 
equal  to  the  velocity  of  approach.    An  imperfectly  elastic  body  is 
one  whose  velocity  of  recoil  is  less  than  its  velocity  of  approach. 
A  perfectly  inelastic  body  is  one  whose  velocity  of  recoil  is  zero. 
The  ratio  of  the  velocity  of  recoil  to  that  of  approach  is  called 
the  coefficient  of  restitution. 

281.  The  principle  of  conservation  of  momentum  is  stated 
in  the  enunciation  of  292.     In  applying  it  to  the  case  sup- 
posed, we  have  for  the  material  system  the  two  equal  bodies  ; 
and  since  their  velocities  are  equal  and  in  opposite  directions, 
the  algebraic  sum  of  their  momenta  is  zero.     The  collision 
does  not  alter  this  sum,  for  it  brings  both  bodies  to  rest. 


LESSON  VII. 

306.  Let  m,  mr  denote  the  masses,  d  their  distance  apart ; 
then,  — 

„  mm) 

force  =  —w~  • 
d? 

The  simplest  unit  of  attractive  force  is  defined  by  putting 

m  =  mf  =  d  =  1. 


160  ELEMENTARY  PHYSICS. 


LESSON  IX. 

335.  Regarding  bodies  as  composed  of  particles,  each,  acted 
on  by  gravity,  Mr.  Stewart  has  given  in  §  47  a  definition  of 
the  centre  of  gravity  of  a  body  which  may  be  stated  in  other 
words  thus  :  —  A  point  in  the  body  such  that,  if  supported, 
the  body  will  remain  at  rest,  but,  if  not  supported,  the  body 
will  fall  under  the  action  of  gravity.     But  this  definition  is 
an  obvious  consequence  of  the  following  mechanical  defini- 
tion, which  should  be  read  in  connection  with  the  solution  of 
195 :  the  centre  of  gravity  of  a  body  is  the  centre  of  the  system 
of  parallel  forces  exerted  by  gravity  upon  the  body. 

336.  A  body  is  said  to  be  homogeneous,  from  a  mechanical 
point  of  view,  when  equal  volumes  of  the  body  have  equal 
mass'es.     A  plane  of  symmetry  in  a  body  is  a  plane  such  that 
every  perpendicular  line  passing  through  it  cuts  the  surface  of 
the  body  in  two  points  equally  distant  from  the  plane.    A  body 
may  have  more  than  one  plane  of  symmetry  ;  in  a  sphere,  for 
example,  every  great  circle  is  a  plane  of  symmetry. 

Similarly,  if  a  line  can  be  drawn  through  a  body,  such  that 
every  perpendicular  drawn  through  the  line  cuts  the  surface  of 
the  body  in  two  points  equally  distant  from  the  line,  this  line 
is  called  a  line  (or  axis)  of  symmetry,  with  respect  to  the  body. 

The  surface  of  the  body  is  said  to  be  symmetrical  with  re- 
spect to  the  plane,  or  the  line,  as  the  case  may  be. 

A  point  is  called  the  centre  of  a  surface  when  every  straight 
line  drawn  through  the  point  and  terminated  by  the  surface 
is  bisected  by  the  point.  The  centre  of  a  surface  is  also  the 
centre  of  the  body  which  is  bounded  by  the  surface.  Considered 
in  this  light,  it  is  also  called  the  centre  of  figure,  or  geometric 
centre. 

340.  In  this,  and  the  three  next  Exercises,  the  student 
should  find  and  state  the  particular  conditions  of  equilibrium 
for  each  case,  and  should  define  and  distinguish  between  stable, 
unstable,  and  neutral  equilibrium,  and  give  examples  of  each 
kind. 

The  general  principle  to  be  applied  in  all  cases  is  that  the 
centre  of  gravity  must  be  supported  in  order  to  have  equi- 
librium. 

348.  The  principal  use  of  the  pendulum  in  Physics  is  to 
determine  the  value  of  the  acceleration  due  to  gravity. 


ANSWERS  AND  SOLUTIONS.  161 

Taking  the  formula  for  the  simple  pendulum  given  in  360, 
we  easily  find  that  g  =  -j^~,  whence  it  follows  that  the  value 

of  g  can  be  found  hy  making  a  pendulum  vibrate,  and  measur- 
ing I  and  T.  But  to  measure  accurately  these  quantities  is 
by  no  means  a  simple  matter. 

In  the  first  place,  the  formula  applies  directly  only  to  the 
simple  pendulum,  which  is  defined  as  a  heavy  particle  sus- 
pended from  a  fixed  point  by  an  inextensible  weightless  thread. 
Such  a  pendulum  can  exist  only  in  the  mind,  and  is  a  concep- 
tion employed  by  the  investigator  for  the  sake  of  simplifying 
the  inquiry  after  the  laws  of  pendulum  oscillations.  In  prac- 
tice we  employ  bodies  which  oscillate  about  a  horizontal  axis, 
called  the  axis  of  suspension ;  these  are  termed,  for  the  sake 
of  distinction,  compound  pendulums.  The  length  of  a  com- 
pound pendulum  is  the  length  of  a  synchronous  simple  pendu- 
lum, that  is,  of  a  simple  pendulum  which  will  oscillate  in  the 
same  time  as  the  given  compound  pendulum.  This  length  can 
be  calculated  in  a  given  case,  when  the  pendulum  is  of  regular 
form,  by  the  aid  of  formulae  which  are  given  in  higher  treatises 
on  Dynamics  ;  but  its  value  is  more  easily  obtained  by  what  is 
called  Kater's  method. 

This  method  is  founded  on  a  remarkable  property  of  a  com- 
pound pendulum,  discovered  by  Huyghens,  called  the  converti- 
bility of  the  axes  of  suspension  and  oscillation.  The  axis  of 
oscillation  is  an  axis  parallel  to  the  axis  of  suspension  in  the 
plane  containing  it  and  the  centre  of  gravity  of  the  pendulum, 
and  at  a  distance  from  it  equal  to  the  length  of  the  synchro- 
nous simple  pendulum.  The  body,  in  fact,  oscillates  as  if  its 
entire  mass  were  collected  on  the  axis  of  oscillation.  Now  the 
property  discovered  by  Huyghens  was  this  :  that  if  we  suspend 
a  pendulum  by  its  axis  of  oscillation,  the  former  axis  of  sus- 
pension becomes  the  new  axis  of  oscillation,  and  the  pendulum 
oscillates  in  the  same  time  as  before.  Kater  constructed  a 
reversible  pendulum,  which  could  be  supported  by  either  of 
two  parallel  knife-edges,  one  of  which  could  be  adjusted  to 
any  distance  from  the  other.  The  length  of  this  pendulum 
could  be  found  by  the  method  of  repeated  trials  with  a  great 
degree  of  accuracy. 

In  order  to  measure  T,  the  method  which  naturally  sug- 
gests itself  is  to  count  the  number  of  oscillations  which  take 


162  ELEMENTARY  PHYSICS. 

place  in  a  given  time,  and  then  divide  the  time  by  the  number 
of  oscillations. 

This  is,  however,  far  from  an  easy  process,  if  accuracy  in 
the  results  is  aimed  at.  Borda  devised  a  great  improvement 
upon  this  method,  by  comparing  the  motion  of  the  pendulum 
with  the  motion  of  the  pendulum  of  an  astronomical  clock 
regulated'  to  beat  seconds.  By  the  use  of  Borda' s  method, 
called  the  "method  of  coincidences,"  one  can  calculate  the 
number  of  oscillations  without  being  obliged  to  count  them. 

Another  method  of  finding,  by  the  aid  of  a  pendulum,  the 
value  of  g  consists  in  employing  a  seconds  pendulum  of  regu- 
lar form  ;  in  this  case  T  —  1,  and  I  can  be  calculated  in  the 
manner  already  stated. 

368.  Let  P  denote  the  force  or  pressure  perpendicular  to 
two  surfaces  in  contact,  by  which  the  surfaces  are  pressed  to- 
gether ;  and  let  F  denote  the  least  force  parallel  to  the  sur- 
faces in  contact  which  is  able  to  move  one  surface  along  the 
other  :  then,  the  ratio  of  Fto  P  is  called  the  coefficient  of  fric- 
tion for  the  two  surfaces. 

If  a  body  be  placed  on  an  inclined  plane  whose  angle  of  in- 
clination can  be  altered  at  pleasure,  then  that  inclination  of 
the  plane  for  which  the  body  is  just  about  to  slide  is  called  the 
angle  of  friction  for  the  materials  of  which  the  body  and  the 
plane  are  composed. 

370.  There  are  five  forces  to  be  considered  :    (1)  Gravity, 
which  may  be  regarded  as  acting  entirely  at  the  centre  of 
gravity  of  the  ladder,  (2)  and  (3)  the  reactions  of  the  floor  and 
wall  perpendicular  to  the  surface  in  each  case,  (4)  and  (5)  the 
forces  due  to  friction,  and  which  act  parallel  to  the  surface  of 
the  floor  and  wall. 

371.  Referring  back  to  Fig.  15,  and  using  the  same  nota- 
tion, it  is  plain  that  when  the  body  is  on  the  point  of  sliding, 
the  friction  F  must  be  equal  to  the  component  of  W9  which 

is  parallel  to  the  plane,  that  is  to  W ,-  — rr-.  The  pressure  R 
on  the  plane  is  equal  to  W -, — — r .  Therefore  we  have 

,,  ~  .          -  f  .  , .  F       height 

the  coefficient  of  friction  =  —  =  y-r3 — . 
It         base 


ANSWERS  AND  SOLUTION'S. 


163 


LESSON  XI. 

376.  A  perfect  fluid  is  an  ideal  conception,  like  that  of  a 
rigid  or  smooth  body  ;  it  is  defined  as  a  body  incapable  of  resist- 
ing a  change  of  shape.  Common  liquids  and  gases  fulfil  this 
definition  when  in  a  state  of  rest,  but  no  existing  fluid  fulfils 
the  definition  when  it  is  in  motion. 

"All  actual  fluids  are  imperfect,  and  exhibit  the  phenome- 
non of  internal  friction  or  viscosity,  by  which  their  motion 
after  being  stirred  about  in  a  vessel  is  gradually  stopped,  and 
the  energy  of  the  motion  converted  into  heat."  (Maxwell.} 

A  viscous  fluid  is  a  substance  such  that  the  very  smallest 
force  applied  to  it  will  produce  a  constantly  increasing  change 
of  form.  The  change  of  form  may  take  place  very  slowly  ;  but 
if  it  takes  place  so  as  to  be  sensible,  and  continually  increases 
with  the  time,  the  substance  is  viscous. 

"Thus  a  block  of  pitch  may  be  so  hard  that  you  cannot 
make  a  dint  in  it  by  striking  it  with  your  knuckles  ;  and 
yet  it  will,  in  the  course  of  time,  flatten  itself  out  by  its 
own  weight,  and  glide  down  hill  like  a  stream  of  water." 
(Maxwell.} 

378.  It  follows  from  the  foregoing  definition  of  a  perfect 
fluid,  that  its  pressure  on  any  surface  must  be  at  all  points 
perpendicular  to  the  surface.     The  student  should  be  able  to 
give  the  reason. 

379.  Let  S,  S',  denote  the  two 
surfaces,  and  P,  P1,  the  pressures  ; 
then   S  :  S1  =  P  :  P.       Pascal's 
principle  cannot  be  proved  direct- 
ly, on  account  of  the  action  of  the 
force  of  gravity. 

385.  The  proof  required  is  sim- 
ply this  :  if  the  resultant  of  all 
the  forces  which  act  on  the  liquid 
at  any  point  of  its  surface  were 
not  normal  (i.  e.  perpendicular)  to 
the  surface  at  that  point,  then  re- 
solve the  resultant  into  compo- 
nents perpendicular  and  parallel 
to  the  surface.  The  first  compo- 
nent is  neutralized  by  the  resist- 


FIG.  18. 


164 


ELEMENTARY  PHYSICS. 


ance  to  compression  of  the  liquid,  liquids  being  practically 
incompressible  ;  the  other  component  would  cause  the  particle 
on  which  it  acts  to  move  along  the  surface,  which  is  contrary 
to  the  supposition  that  the  liquid  is  at  rest. 

Thus,  in  Fig.  18,  let  A  C  represent  this  resultant.  Then 
A  D  and  A  B  are  the  components  referred  to,  and  it  is  easy 
to  see  that  the  component  A  B  being  unresisted  would  cause 
motion. 

386.  The  surface  is  everywhere  termed  horizontal.  Small 
extents  of  the  earth's  surface  may  be  considered  planes,  but 
large  areas  must  be  regarded  as  nearly  spherical. 

387.  (1)  follows  from 
Pascal's    principle,    (2) 

,  _  is  self-evident,  and  (3) 

is  likewise  self-evident 
when  the  meaning  of 
density  is  taken  into 
account. 

388.  By   387    the 
pressure  is  proportion- 
al to  area  X  depth  X 
density.      Now  if  the 
centimetre  be  used  as  a 
unit    of    length,     this 
product   is   by  22  the 
weight  in  grammes  of  a 
volume  of   the  liquid, 
having  the   given  area 
and  depth. 

Adopting  the  unit  of 
weight  as  the  unit  of 
hydrostatic  pressure, 
this  product  is  the 
pressure  in  grammes. 

399.  Let  d,  d',  de- 
note the  densities  of 
the~  two  liquids,  d)  be- 
ing greater  than  d.  Let 
C  D  and  A  B  (Fig. 
19)  be  horizontal  lines 
FIG.  19.  drawn  through  the  sur- 


A/I 


B 


ANSWERS  AND  SOLUTIONS. 


165 


faces  of  the  two  liquids  at  D  and  B,  the  surface  of  the  lighter 
liquid  being  at  D.  Let  M  N  be  a  horizontal  line  drawn 
through  the  common  surface  of  contact  of  the  liquids  at  S, 
and  let  C  A  M  be  a  vertical  line.  Finally,  let  s,  s1,  denote  the 
areas  of  sections  of  the  tube  at  S  and  N  respectively,  and  put 
C  M  =  h,  and  A  M  =  h1.  Since  the  liquid  columns  are  in 
equilibrium,  it  follows  from  387  (1)  that  the  pressures  ex- 
erted upon  the  surfaces  s  and  sf  by  the  superincumbent  liquids 
must  be  as  the  areas  of  the  surfaces.  The  pressure  upon  s  = 
slid ;  that  upon  sf  =  s1  h'  dt.  Therefore 


shditfh'd*  =s: 

h  d  =  h'  df 
h  :  h'  =  d'  :  d 


or 
that  is, 

400.  The  pressures  against  the  various  points  on  the  ver- 
tical side  of  the  vessel  form  a  system  of  parallel  forces,  each 
force  being  proportional  to  the  depth  of  the  point  to  which  it 
is  applied.  This  condition  of  things  is  partially  represented 
by  the  arrows  drawn  in  Fig.  20. 


FIG.  20. 


166  ELEMENTARY  PHYSICS. 

A  very  little  reflection  is  sufficient  to  show  that  the  point 
of  application  of  the  resultant  of  this  system  must  be  some- 
where below  G,  the  centre  of  gravity  of  the  side.  This  be- 
comes evident  by  considering  that  all  the  pressures  below  G 
are  greater  than  any  of  the  pressures  above  G.  Calculation 
shows  that  this  point  of  application  for  a  rectangular  surface 
is  at  a  point  C,  just  two  thirds  the  depth  of  the  side.  This 
point  is  called  the  centre  of  pressure  of  the  side. 

405.  For  the  definition  and  measure  of  density,  see  Solution 
of  21.  The  specific  gravity  of  a  substance  is  the  ratio  of  its 
density  to  that  of  some  standard  substance,  usually  water  at  its 
maximum  density. 

The  density  of  water  in  the  Metric  System  being  unity,  it 
follows  that  in  this  system  the  density  and  specific  gravity  of 
a  substance  are  numerically  the  same. 

422.  The  method  commonly  employed  is  to  attach  to  the 
body  a  sinker,  that  is,  a  body  heavier  than  water,  and  large 
enough  to  cause  both  bodies  to  sink.  The  specific  gravity  of 
the  sinker  being  supposed  known,  it  is  easy  to  determine  that 
of  the  body ;  the  student  should  deduce  a  formula  for  this 
purpose. 

LESSON  XII. 

452.  The  three  forms  referred  to  are  as  follows  : — 

1.  By  the  number  of  statical  units  of  force  in  the  pressure 
on  unit  of  area  ;  e.  g.  in  pounds'  weight  for  square  inch,  or  in 
kilogrammes'  weight  per  square  centimetre.      Pressures  thus 
expressed  are  reduced  to  dynamical  measure  by  multiplying 
by  the  value  of  g  at  the  given  locality. 

2.  By  the  height  of  a  column  of  mercury  at  0°  C.  which 
would  exert  by  its  weight  an  equal  pressure  ;  thus  the  average 
pressure  of  the  atmosphere  is  described  as  a  pressure  of  30 
inches,  or  76  centimetres,  of  mercury. 

3.  In  terms  of  a  large  unit  which  is  nearly  equal  to  the 
average  atmospheric  pressure  at  the  level   of  the  sea.     This 
unit  is  called  an  atmosphere,  and  is  used  chiefly  in  measuring 
pressures  in  boilers,  and  in  scientific  experiments  which  re- 
quire very  great  pressures. 

These  three  measures  are  thus  related  :  in  the  British  sys- 
tem, one  atmosphere  —  pressure  due  to  a  height  of  29 '905 


ANSWERS  AND  SOLUTIONS.  167 

inches  of  mercury  at  32°  F.  at  London,  where  the  force  of 
gravity  is  32*1889  feet  =  about  14f  Ibs.  weight  per  square 
inch.  In  the  Metric  system  one  atmosphere  =  pressure  due 
to  a  height  of  76  centimetres  of  mercury  at  0°  C.  at  Paris, 
where  the  force  of  gravity  is  9*80868  metres  =  about  1033 
grammes'  weight  per  square  centimetre. 

One  British  atmosphere  =  0  '99968  of  a  Metric  atmosphere. 

455.  The  volume  of  the  mercury  resting  upon  unit  section 
of  the  tube  at  the  bottom  of  the  barometer  is  h,  and  its  mass, 
and  also  its  weight,  is  13  '596  h.  This  is  reduced  to  dynamical 
measure  by  multiplying  by  g. 

460.  In  such  questions  as  this  the  effect  of  any  increase  in 
the  atmospheric  pressure  on  the  density  of  water  is  too  small 
to  be  considered. 

464.  Let  <r  =  specific  gravity  of  the  air  at  the  sea-level, 
p  =  that  of  mercury,  h  —  height  of  barometer  at  the  sea-level. 
Then  the  pressure  per  unit  area  indicated  by  the  barometer  is 
h  p.  If  x  =  height  of  the  homogeneous  atmosphere,  the  press- 
ure produced  by  it  per  unit  area  is  x  <r.  Hence,  x  a  =  h  p  ; 

or,  x  =  —  h.     To  find  x  in  metres,  substitute  for  h  76  centi- 
cr 

metres,  for  p  13*596,  and  for  <r  T^T. 

473.  No  account  is  to  be  taken  of  the  effect  of  increased 
pressure  on  the  density  of  the  water. 

478.  Let  V  and  v  denote  the  volumes  of  the  receiver  and 
the  barrel  respectively,  D0  the  initial  density  of  the  air,  D\, 
Z>2,  Ds,  &c.,  ....  Dn,  the  densities  after  1,  2,  3,  &c.,  .  .  .  .  n, 
strokes  of  the  piston.  When  the  piston  is  first  raised  the  vol- 
ume V  of  air  becomes  increased  to  V  H-  v.  Hence  by  Boyle's 
law,  — 

Do:  J)l==  jr+v:  V, 

A  =  A 

Similar  reasoning  shows  that 


and  thus  finally  we  have 


168  ELEMENTARY  PHYSICS. 

479.  One  limit  is  pointed  out  by  Mr.  Stewart  at  the  close  of 
§  93.  Since  each  stroke  of  the  piston  only  removes  a  part  of 
the  air  which  remains,  it  is  plain  that  the  receiver  can  never 
be  completely  exhausted.  This  is  indicated  by  the  formula 
of  the  last  Exercise ;  as  n  increases,  the  value  of  Dn  diminishes 
towards  zero  as  a  limit,  but  this  limit  is  reached  only  when 
n  is  made  greater  than  any  assignable  quantity. 

There  are  other  causes  which  tend  to  produce  an  actual  limit 
after  a  definite  and  not  very  large  number  of  strokes. 

In  the  first  place,  it  is  impossible  to  avoid  the  existence  of  a 
small  space  between  the  bottom  of  the  barrel  and  the  bottom 
of  the  piston,  when  the  latter  is  in  its  lowest  position.  This 
space  has  been  called  untraversed  space  ;  and  it  is  evident  that 
it  contains  a  small  quantity  of  air  at  the  atmospheric  pressure, 
when  the  piston  is  in  its  lowest  position.  "When  the  piston  is 
drawn  up,  this  air  is  rarefied  ;  but  unless  its  tension  becomes 
less  than  the  tension  of  the  air  remaining  in  the  receiver,  no 
air  can  flow  from  the  receiver  into  the  barrel,  and  the  pump 
ceases  to  produce  any  effect. 

A  second  cause  is  leakage,  which  exists  in  the  best-con- 
structed air-pumps,  and  which  increases  rapidly  as  the  tension 
of  the  air  in  the  receiver  and  barrel  diminishes.  As  the  piston 
descends  it  expels  a  certain  quantity  of  air ;  as  it  ascends  a 
certain  quantity  of  air  enters  from  leakage.  If  these  two 
quantities  are  equal,  the  limit  of  rarefaction  has  evidently 
been  reached. 

Lastly,  perhaps  the  most  serious  cause  is  the  absorption  of 
air  by  the  oil  used  in  lubricating  the  piston.  This  oil  is 
poured  upon  the  top  of  the  piston,  where  it  is  forced  by  the 
pressure  of  the  external  air  between  the  piston  and  the  sides 
of  the  barrel,  and  finally  falls  to  the  bottom  of  the  barrel. 
Here  it  absorbs  air  which  it  gives  out  in  part  during  the 
ascent  of  the  piston.  Hence  arises  another  limit  to  the  degree 
of  rarefaction. 


APPENDIX. 

(Containing  useful  Data,  Tables,  and  Formulae.) 
I. 

ENGLISH  WEIGHTS  AND  MEASURES. 

THE  fundamental  units  of  Time,  Length,  and  Mass,  respec- 
tively, are  the  Mean  Solar  Day,  the  Imperial  Standard  Yard, 
and  the  Imperial  Standard  Pound  Avoirdupois. 

The  Mean  Solar  Day  is  the  average  interval  between  two 
successive  passages  of  the  sun  across  the  meridian.  The  mean 
solar  day  is  divided  into  24  hours,  each  hour  into  60  minutes, 
and  each  minute  into  60  seconds,  so  that  one  second  is  s^iinr 
part  of  a  day.  For  a  great  number  of  purposes  the  mean  solar 
day  is  an  inconveniently  large  unit,  and  the  minute  or  second 
is  therefore  employed. 

The  Imperial  Standard  Yard  is,  by  Act  of  Parliament, 
the  distance  between  two  points  in  a  certain  bronze  bar  deposited 
in  the  Office  of  the  Exchequer  in  London,  the  temperature  of  the 
bar  being  62°  F.  (see  18  and  19  Viet.  c.  72,  July  30,  1855). 

The  Imperial  Standard  Pound  Avoirdupois  is,  by 
the  Act  above  cited,  a  platinum  weight  marked  "P.  S.  1844, 
1  lb.,"  deposited  in  the  Office  of  the  Exchequer  in  London. 

TABLE  I. 

Measures  of  Length. 


in. 

ft. 

yd. 

rd. 

fur. 

m. 

Inch 

1 

Foot 

12 

1 

Yard 

36 

3 

1 

Rod 

198 

16J 

4 

1 

Furlong 

7960 

660 

220 

40 

1 

Mile 

63  360 

5  280 

1  760 

320 

8 

1 

170 


APPENDIX. 


NOTES.  1.  In  scientific  investigations  the  foot  and  the  inch  are  gener- 
ally employed,  being  more  convenient  than  the  yard.  The  inch  is  sub- 
divided decimally  and  also  binarily  (i.  e.  into  halves,  quarters,  eighths, 
etc.). 

2.  The  mile  in  the  above  table  is  the  statute  mile.    The  geographical 
or  nautical  mile  (also  called  knot)  is  -fa  of  a  degree  of  longitude  on  the 
equator  or  YTETfrTT  °f  the  whole  equator,  and  is  rather  more  than  1'15  stat- 
ute miles. 

3.  A  hand  =  4  inches ;  1  fathom  =  6  feet ;  1  league  =  3  miles. 


TABLE  II. 

Measures  of  Surface. 


sq.  in. 

sq.  ft. 

sq.  yd. 

sq.  rd. 

R. 

A. 

sq.  m. 

Square  Inch 

1 

Square  Foot 

144 

1 

Square  Yard 

1296 

9 

1 

Square  Rod 

39204 

272* 

30i 

1 

Rood 

1  568  160 

10890 

1210 

40 

1 

Acre 

6  272  640 

43560 

4840 

160 

4 

1 

Square  Mile 

4  014  489  600 

27  878  400 

3  097  600 

102  400 

2560 

640 

1 

TABLE  III. 

Measures  of  Volume. 


cub.  in. 

cub.  ft. 

cub.  yd. 

Cubic  Inch 

1 

Cubic  Foot 

1  728 

1 

Cubic  Yard 

46  656 

27 

1 

NOTE.    16  cubic  feet  of  wood  make  1  cord  foot,  and  8  cord  feet'  or  128 
cubic  feet  make  1  cord. 


APPENDIX. 


171 


TABLE  IY. 
Measures  of  Mass  (Avoirdupois  Weights}. 


gr- 

dr. 

oz. 

Ib. 

qr. 

cwt. 

T. 

Grain 

1 

Drachm 

27-34375 

1 

Ounce 

437'5 

16 

1 

Pound 

7  000 

256 

16 

1 

Quarter 

196  000 

7  168 

448 

28 

1 

Cwt. 

784  000 

28  672 

1  712 

112 

4 

1 

Ton 

15  680  000 

573  440 

34  240 

2240 

80 

20 

1 

NOTES.  1.  The  pound  is  connected  with  the  unit  of  volume  as  fol- 
lows :  1  cubic  inch  of  distilled  water  weighed  in  air  at  62°  F.  (bar.  30 
inches)  =  252*458  grains. 

2.  The  Avoirdupois  Pound  of  matter  is  equal  to  the  mass  of  27'7274 
cubic  inches  of  distilled  water,  weighed  as  above. 

3.  For  many  purposes  it  is  sufficiently  accurate  to  call  the  weight  of  1 
cubic  foot  of  water  1000  ounces. 


Three  other  measures  are  much  in  use,  viz.,  Liquid 
Measure,  the  base  of  which  is  the  Imperial  Gallon  ;  Dry 
Measure,  the  base  of  which  is  the  Imperial  Bushel  ;  and  Troy 
Weight,  the  base  of  which  is  the  Troy  Pound. 

The  Imperial  Gallon  is  a  measure  of  volume  or  capacity 
which  wTill  contain  10  pounds,  avoirdupois  weight,  of  distilled 
water,  weighed  in  air,  at  62°  F.,  the  barometer  at  30  inches. 

The  Imperial  Bushel  is  also  a  measure  of  capacity,  and  is 
equal  to  8  imperial  gallons. 

The  Troy  Pound  is  a  measure  of  mass,  bearing  to  the 
Avoirdupois  Pound  the  ratio  of  5760  :  7000  ;  i.  e.  it  contains 
5760  grains'  weight  of  matter. 


172 


APPENDIX. 


TABLE  V. 
Measures  of  Capacity. 


Liquid  Measure. 

Dry  Measure. 

4  gills       =  1  pint  (pt.) 
2  pints      =  1  quart  (qt.) 
4  ruarts    —  1  gallon  (gal.) 
52^  gallons  =  1  hogshead. 

2  pints    =  1  quart. 
8  quarts  =  1  peck  (pk.) 
4  pecks  =  1  bushel  (bush.) 

NOTES.  1.  Liquid  Measure  is  used  in  measuring  liquids,  and  Dry 
Measure  in  measuring  solid  matter  consisting  of  small  parts  or  pieces,  as 
grain,  fruit,  salt,  roots,  ashes,  &c. 

2.  The  tun  =  2  pipes  =  4  hogsheads  =  210  Imperial  gallons. 


TABLE  VI. 
Troy  Weights. 


gr. 

dwt. 

oz. 

Ib. 

Grain 

1 

Pennyweight 
Ounce 

24 

480 

1 

20 

1 

Pound 

5  760 

240 

12 

1 

NOTES.     1.  Troy  weight  is  chiefly  employed  in  weighing  gold,  silver, 
and  precious  stones. 

2.  Apothecaries,  in  compounding  medicines,  divide  the  ounce  (3)  into 
8  drachms  (5)  and  the  drachm  into  3  scruples  O),  so  that  1  scruple  =  20 
grains. 

3.  480  minims  =  1  fluid-ounce,  20  fluid-ounces  =  1  pint. 


APPENDIX.  173 

II. 

UNITED  STATES  WEIGHTS  AND   MEASURES. 

THE  fundamental  unit  of  time  in  the  United  States,  as  in 
England,  is  the  mean  solar  day.  It  is  also  divided,  as  in  Eng- 
land, into  hours,  minutes,  and  seconds. 

The  United  States  standards  of  length  and  mass  are  copies 
of  old  English  standards,  arid  are  very  nearly  the  same  as  the 
present  Imperial  standards  of  Great  Britain. 

Careful  comparison  has  shown  that  the  United  States 
actual  standard  yard  (a  brass  scale  made  by  Troughton  of 
London,  in  1813,  and  deposited  in  the  Office  of  Weights  and 
Measures  in  Washington)  is  equal  to  1  '000024  Imperial  yards, 
and  that  the  United  States  actual  standard  of  mass  (a  Troy 
pound  of  brass,  made  by  Kater  in  1827,  and  deposited  in  the 
United  States  Mint)  is  equal  to  0 '99999986  Imperial  Troy 
pounds.  The  same  ratio  0*99999986  also  exists  between  the 
United  States  and  Imperial  Avoirdupois  pounds. 

The  United  States  Gallon,  or  standard  unit  of  liquid 
measure,  is  the  old  wine  gallon  of  England,  and  contains  231 
cubic  inches. 

The  United  States  Bushel,  or  standard  unit  of  dry 
measure,  is  the  British  Winchester  bushel  (formerly  a  standard 
in  England),  and  contains  21 50 '42  cubic  inches. 

The  relative  value  of  the  Imperial  and  United  States  stand- 
ards of  capacity  is  as  follows  :  — 


Comparative  Values  of  English  and  TJ.  S.  Units  of  Capacity. 


1  Imperial  Gallon  (277-274  cub.  in.)  =  1-2001    U.  S.  Gallons. 
1  U.  S.  do.      (231-000  cub  in.)  =  0-8331    Imperial  Gallons. 

1  Imperial  Bushel  (2218-192  cub.  in.)  =  1-0315    U.  S.  Bushels. 
1  U.  S.  do.     (2150-420  cub.  in.)  =  0-96945  Imperial  Bushels. 


The  ratio  of  the  United  States  and  Imperial  gallons  is  very 
nearly  as  5  :  6  ;  and  that  of  the  two  bushels  nearly  as  16  :  17. 

With  one  exception,  the  United  States  tables  of  Weights  and 
Measures  are  the  same  as  the  English  tables  which  have  been 


174  APPENDIX. 

given.  This  exception  occurs  in  Avoirdupois  Weight,  in  which 
the  United  States  quarter  is  equal  to  25  Ibs.,  and  the  United 
States  ton  therefore  is  equal  to  2000  Ibs. 

NOTES.  1.  The  English  (long  or  gross)  ton  of  2240  Ibs.  is  still  used  in 
estimating  English  goods  in  the  United  States  Custom  Houses,  in  selling 
coal  at  wholesale  from  the  Pennsylvania  mines,  and  in  the  wholesale  iron 
and  plaster  trade. 

2.  In  liquid  measure  63   United    States  gallons  =  52|  Imperial  gal- 
lons =  1  hogshead. 

3.  1  United  States  fluid-ounce  =  1  fluid-ounce  and  20  minims,  Imperial 
measure  ;  and  16  United  States  fluid-ounces  =  1  United  States  pint. 

4.  The   United  States  barrel    contains  from  28  to  32   United  States 
gallons. 

5.  In  dry  measure  the   half-peck,    sometimes  called  the  dry  gallon, 
contains  268' 8  cubic  inches,  so  that  liquid  and  dry  measures,  of  the  same 
name  (for  example,  the  quart),  stand  to  each  other  in  value  as  231  :  268*8. 

6.  'The  bushel,   heaped  measure,  contains  about  2750  cubic  inches,  or 
rather  more  than  5  pecks. 


APPENDIX.  175 

III. 

THE  METRIC  SYSTEM  OF  WEIGHTS  AND  MEASURES. 

The  fundamental  units  of  Time,  Length,  and  Mass,  respec- 
tively, are  the  mean  solar  day,  the  metre,  and  the  kilogramme. 

The  Mean  Solar  Day  has  been  already  denned  (see  Eng- 
lish Weights  and  Measures). 

All  civilized  nations,  in  fact,  employ  the  mean  solar  day  as 
the  fundamental  unit  of  time,  and  divide  it  into  hours,  min- 
utes, and  seconds,  as  in  England  and  the  United  States. 

Its  universal  use  is  due  to  the  fact  that  it  combines,  in  an 
unrivalled  degree,  the  cardinal  virtues  of  a  fundamental  unit 
of  measurement.  Its  magnitude  is  determined,  not  by  legis- 
lators or  men  of  science,  but  by  nature ;  its  duration  can  be 
easily  ascertained  to  a  high  degree  of  precision  ;  it  has  not 
changed  since  the  time  of  Hipparchus  (150  B.  c.)  by  so  much 
a§  TOTT  of  a  second  ;  and  finally,  the  purposes  of  life  and  the 
actions  of  men  are  so  largely  dependent  upon  the  position  of 
the  sun  that  a  measure  of  time  derived  from  his  motion  is 
vastly  more  convenient  than  any  other. 

The  Metre,  or  standard  of  Length,  is  the  distance  between 
the  ends  of  a  certain  platinum  bar  made  by  Borda,  and  pre- 
served in  the  Archives  de  1'Etat  in  Paris,  the  bar  being  at  the 
temperature  of  melting  ice.  The  metre  was  intended  to  be  an 
exact  ten-millionth  part  of  a  quadrant  of  the  earth's  meridian, 
but  according  to  the  most  trustworthy  measurements  the  actual 
standard  metre  ( Borda' s  platinum  rod)  is  less  than  its  intended 
value  by  an  amount  not  exceeding  oVo  of  a  metre,  or  about 
y^  of  an  inch. 

The  Kilogramme,  or  standard  of  Mass,  is  the  quantity  of 
matter  in  a  certain  platinum  weight  made  by  Borda,  and  depos- 
ited in  the  Archives  de  1'Etat  in  Paris.  It  was  intended  to  be 
(and  is  very  nearly)  equal  to  the  mass  of  one  cubic  decimetre  of 
distilled  water  at  the  point  of  maximum  density  (about  4°  C.). 

The  metre  and  the  kilogramme  derive  their  authority  as 
standards  from  a  law  of  the  French  Republic  in  1795.  Of  the 
prototype  standards,  kept  at  Paris,  numerous  copies  have  been 
taken,  which,  after  having  been  compared  with  the  originals 
with  the  utmost  precision  of  which  modern  science  is  capable, 
have  been  made  standards  of  reference  and  verification  in  the 
various  countries  ift  which  the  Metric  System  has  been  adopted . 
5* 


176  APPENDIX. 

The  Metric  System  of  Weights  and  Measures  is  a  system  in 
the  true  sense  of  that  word,  and  the  simple  mode  in  which  all 
the  other  units  are  derived  from  the  standards  is  in  striking 
contrast  with  the  want  of  system  and  troublesome  numerical 
relations  existing  in  English  Weights  and  Measures. 

Nearly  all  the  derived  units  in  the  Metric  System  are  obtained 
by  the  application  of  two  very  simple  principles,  viz.  decimal 
multiplication  and  division,  and  the  use  of  squares  and  cubes 
of  linear  units  as  units  of  surface  and  of  volume  respectively. 
Hence  each  unit  of  length,  and  also  of  mass,  is  10  times 
larger  than  the  next  smaller  unit  in  order,  each  unit  of  surface 
100  times  larger,  and  each  unit  of  volume  1000  times  larger. 

Names  for  all  the  units  are  formed  from  the  roots  metre  and 
gramme,  by  employing  the  Greek  prefixes,  deca  (]  0),  hecto  (100), 
kilo  ,(1000),  for  the  multiples,  and  the  Latin  prefixes,  deci  (O'l), 
centi  (0*01),  milli  (O'OOl),  for  the  sub-multiples.  In  the  meas- 
ures of  surface  and  of  volume  the  words  square  and  cubic  re- 
spectively are  also  employed.  Thus,  1  centim-etre  is  equal  to 
O'Ol  of  a  metre,  1  cubic  decimetre  is  equal  to  0*001  of  a  cubic 
metre,  1  kilogramme  is  equal  to  1000  grammes,  &c. 
Four  units  have  received  distinct  names,  viz.  :  — 

The  Are,  which  is  equal  to  1  square  decametre. 

The  Stere,  which  is  equal  to  1  cubic  metre. 

The  Litre,  which  is  equal  to  1  cubic  decimetre. 

The  Tonne,  which  is  equal  to  1000  kilogrammes. 
The  table  opposite  exhibits  the  names,  abbreviations,  and 
relative  values  of  nearly  all  the  measures  in  the  Metric  System. 

NOTES  1.  The  Are  is  employed  as  a  unit  of  land  measure  ;  the  Stere 
as  a  unit  for  measuring  cord- wood  ;  and  the  Litre  as  a  unit  for  measuring 
fluids,  and  dry  substances  in  small  pieces,  as  grain,  fruit,  salt,  &c. 

2.  Decimal  multiples  and  divisions  of  the  litre  are  in  use,  and  are 
named  by  employing  Greek  and  Latin  prefixes  in  the  same  way  as  in 
measures  of  length ;  thus,  decilitre,  kilolitre,  &c.     Similarly,  100  ares  is 
called  a  hectare,  YQTT  °f  an  are  a  centiare,  lOsteres  a  decastere,  -^o  of  a 
stere  a  decistere,  10000  metres  a  myriametre,   10000  grammes  a  myria- 
gramme. 

3.  Binary  derivatives,  as  in  England,  are  much  employed  on  account 
of  their  practical  convenience ;   e.  g.   the  demi-litre,   the  double  hecto- 
gramme, &c.,  &c. 

4.  The  cubic  metre,  when  employed  in  France  as  a  measure  of  the 
capacity  of  ships,  is  called  a  tonneau.    The  word  tonneau  also  denotes, 
in  the  French  navy,  the  weight  of  a  cubic  metre  of  water. 

5.  100  kilogrammes  is  called  a  quintal. 


APPENDIX. 


177 


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g        fl        05      >^       ^        g        o> 

i 

S 

1  i  1:-*  1  1  1 
111*111 

3 

1  1  1  1  1  1  1 

O 
O 

o 

T—  1 

II 

* 

11 

fi 

IT  i  a  •     IT  a  i 

o 
EH 

^       ^3-      r§                       S         0          g 

1 

I 

f  i^  s^5^ 

ii  it  us 

i 

^ 

,3      o      o     ±?      o      g     i—  i 

§ 

fr 

•  rH           O            <D           Zi           OJ           <JJ         "S 

^     ,d     -^      S     nd      o      S 

555 

o 

0-0        0        O        0        0        0 

S 

i 

S3       3       d       2       S3       0       S 

^ 

P3 

O    O    O    Q    O    O    Q 

'•I 

| 

*      * 

1 

1  1  1     }  .  f  f 

"S 

Surface. 

H 

iiifiii 

column  dent 
it. 

35        rS        ^          fi        'S          0          S 

1] 

<V        O        <D        CD        <D        g        O 

cS       c8       o3       o3       c^       «3       c3 

®*  n 

J*  I1  ti*  J1  If  w*  ^ 

|| 

^1?    7%^ 

S  1 

ii 

1 

0 

M    v—^  "^    *     "^    ^s    B 

^^  x    g       "      •    s  —  '    <jj 

Is-S 

11 

I 

H 

1  1  !  5  1  1  1 

II 

^O      o      o      q)      o      j^j     ^ 

^  'e 

3  W""Q  S  Q  o  S 

•*- 

178 


APPENDIX. 


(M  CO  rH  J>-  OS 

CO  CO  rH  !>.  "*  CM  O  t^COC 

-*  OS  CO  CO  OOOrH  rH  CO  < 

O  CO  CM  *—  rH  OS  CO  CO  O  C 

t^  OO  CO  CO  -Hi  O  OO  VQ  OS  C 

TO  O  CO  rH  CO  r— I  CO  r- 1  1O  < 

OS  CO  OS  C^l  J>-  t>«  O  CO  J>-  C 

CO  <M  O  co  '-^O  jV,^^ 


)        «0 
5          rH 

•)   CO   00 


O  CO  rH  O      rH  O3  O 


)  rH  O       rH  (N  C 


a     I 
2«-§l  «  j 

cc    <D    2    _i      ID         T-I 

U-s  ^  t    3 

a    r3r31x)        %  £ 

W  2  -22  «    2  g  S 

duu-3  Is  ^ 

^   B   S   fl     *   c^   5a 

S^^'S  .S  d  a 
-e  a  a  S    c*3^ 


II 


g 

1 


IIS 


1>»  OS  OS  C<1      O  J>»  ^      rH  1O  OO 

OSt^COCO      OSCOO5      CO-*iO 

CO  O  rH  O   OS  O  J 


1O  1O  1O 
OS  CO  OS 
OO  C<1  rH 


COO5CO   '—'"^JS   OCO  *O  l>-  "   rH  OS  rH   O  "*  CO 
(N  O  O  rH   Q  O  d   O  CO  ^H  CO  CO  rH  Q  rH   O  O  O 


s     1 

II  I. 

^      S       r2 


a  a  s  fl 

__    •'-'    D  C  o3  M 

^  5i  S  rs  *-jj  -^  o 

o  a  p)  fl  rt  3  ;*sS 

11  CD  f^  rO 


8, 8,:  4 


APPENDIX.  179 

IV. 

MATHEMATICAL  DATA  AND  FORMULAE. 

ALGEBRA. 
Arithmetical  Progression. 

(1)  Let  a  denote  the  first  term,   I  the  last  term,  d  the  com- 
mon difference,  n  the  number  of  terms,  s  the  sum  of  the 
terms ;  then,  — 

I  =  a  +  (n  —  1)^;  s  =  n  — - — . 

2 

Geometrical  Progression. 

(2)  Let  the  same  notation  as  above  be  used,  except  that  in 
this  case  r  denotes  the  common  ratio  ;  then,  — 

rl  —  a 
I  =*  ajTn—\.  s  — 


r  —  l  * 


GEOMETRY. 


(3)  TT  (ratio  of  circumf  'nee  of  circle  to  diameter)  =  3*14159 

(4)  Circumference  of  a  circle  (radius  r)  =  2  irr 

(5)  Area  of  a  circle  (radius  r)  =  TT  r2 

(6)  Surface  of  a  sphere  (radius  r)  =  4  TT  r2 

(7)  Surface  of  ellipse  (semi-axes  a  and  b)  =  TT  a  b 

(8)  Surface  of  right  cylinder  (height  h,  base  TT  r2)  =  2  TT  rh 

(9)  Surface  of  r,ight  cone  (height  h,  base  irr2)     =  TT  r  y/r2  +  h2 

(10)  Volume  of  a  sphere  (radius  r)  =  |  TT  r3 

(11)  Volume  of  ellipse  (semi-axes  a  and  b)         =  f  TT  a  b  c 

(12)  Volume  of  right  cylinder  (height  h,  base  irr2)  —  wr^h 

(13)  Volume  of  right  cone  (height  h,  base  TT  r2)  =  5  irr'2  h 

(14)  Sum  of  the  three  angles  of  a  triangle          =  180° 

(15)  Area  of  a  triangle  of  altitude  h  and  base  b  =  %bh. 

(16)  Two  triangles,   which  have  equal  bases  and  equal  alti- 

tudes, are  equal. 


180  APPENDIX. 

(1 7)  Two  triangles  are  similar  ; 

(a)  If  they  are  equiangular  with  respect  to  each  other. 

(b)  If  they  have  their  homologous  sides  proportional. 

(c)  If  they  have  an  angle  of  one  equal  to  an  angle  of  the 

other,  and  the  sides  including  these  angles  propor- 
tional. 

(18)  The  perpendicular  upon  the  hypothenuse  of  a  right  tri- 

angle from  the  vertex  of  the  right  angle,  divides  the  tri- 
angle into  two  triangles  which  are  similar  to  each  other 
and  to  the  whole  triangle. 

(19)  If  a  perpendicular  be  erected  from  any  point  on  the  di- 

ameter of  a  circle  meeting  the  circumference  at  a  point 
A,  and  chords  be  drawn  from  A  to  the  extremities  of 
the  diameter  ;  then,  — 
•  (a)  The  perpendicular  is  a  mean  proportional  between  the 

segments  of  the  diameter. 

(&)  Either  chord  is  a  mean  proportional  between  the  di- 
ameter and  the  adjacent  segment. 

(20)  The  square  described  upon  the  hypothenuse  of  a  right 

triangle  is  equivalent  to  the  sum  of  the  squares  de- 
scribed on  the  other  two  sides. 

(21)  Similar  triangles  (or  polygons)  are  to  each  other  as  the 

squares  of  their  homologous -sides. 

(22)  The  sum  of  the  plane  angles  which  form  a  solid  angle  is 

always  less  than  four  right  angles. 

(23)  The  area  of  the  surface  described  by  a  line  revolving 

about  another  line  on  the  same  plane  as  an  axis  is  the 
product  of  the  revolving  line  by  the  circumference  de- 
scribed by  its  middle  point. 


PLANE  TRIGONOMETRY. 
Definitions  of  the  Functions  of  an  Angle. 


Let  Ti  denote  the  hypoth- 
enuse of  a  right  triangle,  a  the 
perpendicular,  b  the  base,  A 
and  B  the  angles  opposite  a 
and  b  respectively  ;  then,  — 


APPENDIX. 


A  W 

sin  A  =  T  , 


,  A  w 

tan  .4  =  -  , 


(24) 


(25) 


sec  ^4  =  7  , 
o 


. 

COS  A  =  7  , 

h 


cot  ^4  =  -  , 

Ob 


181 

h 
I* 

h 


cosec  A  =  - . 
Values  of  the  Functions  of  particular  Angles. 


Angle. 

Arc. 

sin 

cos 

tan 

cot 

sec 

cosec 

0° 

0 

0 

1 

0 

GO 

1 

OO 

30° 

JIT 

1 

i/3 

VI 

V/1J 

2  V/l 

2 

45° 

JT 

V/l 

v/F 

1 

1 

vd 

VT2 

60° 

1* 

1V3 

i 

V/l 

V/I 

2 

2  V/l 

90° 

1» 

1 

0 

OO 

0 

OO 

1 

120° 

§T 

isAs 

-i 

-Vi 

-VI 

—  2 

2  V/l 

135° 

i* 

v/1 

-VI 

—  1 

—i 

-V/l 

V'lJ 

150° 

l» 

i 

-i\^ 

-v/1 

-V^ 

-2  V/l 

2 

180° 

7T 

0 

^ 

0 

OO 

—  1 

00 

Useful  Formulae,  in  which  a  and  /3  denote  any  two  Angles. 
(26)  sin2  a  +  cos2  a  =  1. 

sin  a  1 

tan  a  — =  — : — . 

cos  a        cot  a 

sec2  a  =  1  +  tan2  a. 
cosec2  a  =  1  -f  cot2  a. 


(27) 


(28) 
(29) 
(30) 


versed  sin  a  =  1  —  cos  a. 


182  APPENDIX. 

(31)  sin  2  a  =  2  sin  a  cos  a. 

(32)  cos  2  a  =  cos2  a  —  sin2  a. 


(33) 


-  tan2 


(34)  sin  (a  ±  /3)  =  sin  a  cos  /3  ±  cos  a  sin  (3. 

(35)  cos  (a  ±  /3)  =  .cos  a  cos  )S  q=  sin  a  sin  /3. 

x  ,  tan  a  ±  tan  /? 

to(a±flg 


(37) 


cot  £  ±  cot  a 


(38)  '    sin  a  +  sin  j3  =  2  sin  J  (a  -f  /3)  cos  |  (a  —  /3). 

(39)  sin  a  —  sin  f3  =  2  cos  |  (a  +  £)  sin  |  (a  —  /3). 

(40)  cos  a  +  cos  j8  —  2  cos  |  (a  -j-  £)  cos  |  (a  —  /3). 

(41)  cos  a  —  cos  (3  =  —  2  sin  \  (a  -f  p}  sin  |  (a  —  /3). 

In  any  plane  triangle,  let  a,,  b,  c,  denote  the  sides,  A,  B,  C, 
the  angles  respectively  opposite  the  sides,  and  let  5  = 
|  (a  +  b  4-  c)  ;  then,  — 

sin  A  =  smB  =  sin  0' 

(43)  a  +  b  =  tanH^  +  ^)t 

^   a2  =  b2  4-  c2  —  2  b  c  cos  ^4, 

(44)  <    b2  =.  c2  -f-  a2  —  2  c  a  cos  Bf 
(  c2  =  a2  -f-  52  —  2  a  &  cos  (7. 


(45) 


(46) 


(47) 


APPENDIX.  183 

ANALYTIC  GEOMETRY. 
General  Equation  of  the  Straight  Line. 

(48)  Ax  +  By+C==Q. 

In  the  following  equations  of  the  straight  line,  a  and  b  de- 
note the  intercepts  of  the  line  on  the  axes  of  x  and  y  respective- 
ly, m  the  tangent  of  the  angle  which  the  line  makes  with  the 
axis  of  #,  p  the  perpendicular  on  the  line  from  the  origin,  a 
the  angle  which  this  perpendicular  makes  with  the  axis  of  x. 

(49)  1  +  1  =  1. 

(50)  y  =  m  x  -f-  b. 

(51)  x  cos  a  +  y  sin  a  =  p. 

(52)  Two  straight  lines,  the  equations  of  which  are  y  =  m  x 

-{-  b,  and  y  =  m1  x  +  &',  are  parallel  when  m  =  m1, 
and  are  perpendicular  when  m  mf  -f-  1  =  0. 

General  Equation  of  a  Conic  Section. 

(53)  aa2  +  2Az?/  +  &7/2  +  2^  +  2/y  +  c  =  0. 

If  &2  —  a  b  <  0,  locus  is  an  ellipse, 
Ifh2  —  a  b  <  0,  and  a  =  b,  locus  is  a  circle, 
If  h2  —  a  b  >  0,  locus  is  an  hyperbola, 
Ifh2  —  a  b  =  0,  locus  is  a  parabola. 

In  the  following  equations  of  the  conic  sections,  r  denotes 
the  radius  of  the  circle,  a  and  &  the  semi-axes  of  the  ellipse  or 

[~^2         fr2 

hyperbola,  e  =  .!  -— -^ — ,  and  is  the  measure  of  the  eccen- 
tricity of  the  ellipse  or  hyperbola,  p  =  the  parameter  of  the 
parabola  (that  is,  the  double  ordinate  at  the  focus),  m  the 
distance  from  the  vertex  of  a  parabola  to  the  focus  (or  focal 
distance). 

Circle. 

(54)  Centre  at  point  (a  )8)  ;  (x  —  a)2  +  (y  —  p)2  =  r2. 

(55)  Origin  at  centre  ;  x2  +  y2  =  r2. 

(56)  Axis  of  x  diameter,  origin  on  circumf  nee  ;  x2  +  y2  =  2rx. 

(57)  Tangent  at  point  (xf  yf),  origin  at  centre ;  x  x1  -f-  y  y'  =  r2. 


184  APPENDIX. 

Ellipse. 

(58)  Origin  at  centre — 2  +  ^  =  1. 

(59)  Origin  at  vertex y1  =  -^  (2  a  a;  —  x2). 

(60)  Pole  at  centre  ,    p2  = \ =-r 

1  —  e2  cos2  0 

(61)  Pole  at  focus p  =  - 


-\-  e  cos  0)  ' 

(62)  Tangent  at  point  ^  y ~+  ^  -  1. 

(63)  Focal  radii  make  equal  angles  with  the  tangent. 

Hyperbola. 

The  equations  of  the  hyperbola  are  the  same  as  those  of  the 
ellipse,  except  that  b2  is  negative. 

Parabola. 

(64)  Origin  at  vertex y2  =  p  x  —  4  m  x. 

2  IYI 

(65)  Pole  at  focus  .  p  = . 

1  —  cos  0 

(66)  Tangent  at  point  xf  yf     .     .     .     .     2  y  yf  =  p  (x  +  x'). 

(67)  The  subtangent  is  bisected  at  the  vertex. 

(68)  The  subnormal  is  constant  and  equal  to  \p. 

(69)  The  point  where  any  tangent  cuts  the  axis  of  x,  and  its 

point  of  contact,  are  equally  distant  from  the  focus. 

(70)  Any  tangent  makes  equal  angles  with  the  axis  of  x  and 

the  focal  radius  vector. 

Analytic  Geometry  of  Three  Dimensions. 

If  I,  m,  n,  denote  the  cosines  of  the  three  angles  which  a 
straight  line  makes  with  three  rectangular  axes,  or  direction 
cosines  ;  then,  — 

(71)  I'2  +  m2  -f  n2  =  1. 

If  0  denote  the  mutual  inclination  of  two  lines  (I  m  n) 
(I1  m1  nf)  ;  then,, — 

(72)  cos  0  =  I  V  +  m  m'  +  n  n'. 


APPENDIX. 


185 


V. 

PHYSICAL  DATA  AND  TABLES. 
TABLE  I. 

Value  of  the  Acceleration  of  Gravity. 


Place. 

Latitude. 

Value  of  g 
in  Metres. 

Seconds 
Pendulum 
in  Metres. 

Spitzbergen  

79  49  58  N. 

9-83141 

0-99613 

Stockholm  

59  20  34  " 

9-81946 

0-99492 

Konigsberg  
London 

54  42  12   " 
51  30  48   " 

9-81443 
9-81111 

0-99441 
0*99409 

Paris 

48  50  14  " 

9*80979 

0*99394 

Isle  Rawak    

0     1  34  S. 

9-78206 

0-99113 

Isle  de  France  
Cape  of  Good  Hope 
Cape  Horn  

New  Shetland  

20     9  23   " 
33  55  15   " 
55  51  20   " 
62  56  11   " 

9-78917 
9-79696 
9-81650 
9-82253 

0-99185 
0-99264 
0-99462 
0-99523 

The  value  of  g  is  usually  determined  experimentally  by  pen- 
dulum observations.  It  may  be  calculated  with  sufficient 
accuracy  by  the  following  formula,  — 


g  =  G  (1  —  0-0025659  cos  2  X) 


(\  — 


1-32 


in  which  G-  =  value  of  g  for  the  latitude  45°  =  32*1703  feet 
'•=  9-80533  metres,  r  =  radius  of  the  earth  =  20  886  852  feet 
=  6  366  198  metres,  and  h  —  height  of  the  place  in  feet  or 
metres  above  the  level  of  the  sea. 

TABLE  II. 

Dimensions  of  Dynamical  Units, 
T  denoting  a  time,  L  a  length,  and  M  a  mass. 


Dimension. 

Velocity. 

Acceler- 
ation. 

Momen- 
tum. 

Force. 

Energy. 

L 
T 

L 

~T* 

ML 
T 

ML 

ML2 

T2 

T2 

mass  X  velocity  =  mass  X  acceleration  X  time  =  force  X  time. 

186 


APPENDIX. 


TABLE  III. 

Specific  Gravities  of  Solids  and  Liquids,  referred  to  that  of 
Distilled  Water  at  4°  C.  as  a  standard. 


Metals. 
Platinum,  hammered 

22-060 

Roclcs. 
Granite  2*65  to  2  '75 

Gold, 

19-350 

Sandstone  2  '25  to  2  "65 

Silver,  ^ 
Copper"          " 

10-510 
8-900 

Limestone    2*60  to  2  '70 
Marble                    2  '65  to  2  '75 

Lead,  cast  

.  11-350 

Slate                                     2'84 

Tin,  cast  

.    7-290 

Porcelain  Clay                    2  '21 

Zinc  

.   7-190 

Sand   dry                             1-42 

Iron,  cast 

7  -250 

Iron,  wrought 

7780 

Various  Substances 

Steel,  soft  

.  7*830 

Diamond                            3  '530 

Steel,  tempered  

.  7*810 

Glass   flint                         3  "000 

Brass,  cast   

..  7-820 

Brass,  sheet  

.  8-390 

Sulphur                              2  "086 

Bronze,  statuary  

..  8-950 

Brick                                  2  '000 

Bronze,  gun  metal  .... 

..  8-460 

Coal,  anthracite  1'800 

Aluminum   

..  2-600 

Wax  0-960 

Woods. 

Ice     0-918 

Lignum  Vitae  

..  1-333 

Box,  Dutch  

1-328 

Liquids. 

Box,  French    

0'912 

Mercury   .              .         13*596 

Ebony,  American 

1-331 

Bromine                             2  "966 

Oak,  just  felled 

1-113 

Sulphuric  Acid,  com'cial  1  '841 

Oak,  seasoned  

.  0-743 

Nitric  Acid,            "         1'500 

Mahogany,  Spanish  .. 
Beech    

..  1-063 
..  0-852 

Muriatic  Acid         "         1-200 
Chloroform  1'492 

Ash  

.  0-845 

Milk                                   1  '031 

Maple       .  . 

0-790 

Sea-water                            1'025 

Elm  

..  0-673 

Proof  Spirit  0  '920 

Chestnut 

0-565 

Absolute  Alcohol              0*795 

Red  Pine  

..  0-657 

Sulphuric  Ether  0  '  7  1  5 

White  Pine  

..  0-551 

Olive  Oil  0*915 

Larch    

..  0-530 

Oil  of  Turpentine              0'870 

Cork  

.  0-240 

Naphtha                            0'840 

APPENDIX. 
TABLE  IV. 


187 


Specific  Gravities  of  Gases  and  Vapors,  referred  to  that  of  Dry 
Air  at  the  same  Temperature  and  Pressure  as  the  Standard. 


Oxvsren  .  .  . 

1-106 

Vapor  of  Water 

0-623 

'Hydrogen  

0-069 

Carbonic  Oxide 

0-967 

Nitrogen 

0-972 

Carbonic  Acid  .  . 

1-525 

Chlorine        

2-470 

Ether    

2-570 

Ammonia 

0-590 

Marsh  Gas    

0-555 

Nitrous  Oxide  

1-520 

Coal  Gas    0 

420  to  0-520 

Weight  of  1  litre  of  dry  air  at  0°  C.  and  76  centimetres  press- 
ure at  Paris  =  1  '293  grammes.  Weight  of  1  cubic  foot  of  dry 
air  at  32°  F.  and  29 '905  inches  pressure  at  London  =  565 
grains,  or  about  1  '25  ounces  avoirdupois. 

BAROMETRIC  FORMULAE  FOR  FINDING  DIFFERENCES  OF 
LEVEL. 

In  these  formulae  X  denotes  difference  of  level,  h  height 
of  barometer  at  lower  station  reduced  to  0°  C.,  h'  the  same 
for  higher  station,  t  temperature  at  lower  station,  V  the  same 
for  higher  station  (t  and  t?  being  expressed  in  Fahrenheit 
degrees  in  I.  and  in  Centigrade  degrees  in  II. ),  X  the  latitude 
of  the  stations.  The  atmosphere  is  supposed  to  be  in  a  mean 
hygrometric  condition. 

The  second  and  simpler  value  of  X,  given  in  both  I.  and 
II.,  will  suffice  for  heights  not  exceeding  3300  feet,  or  1000 
metres. 

I.      X(i 


(1  +  0-00256  cos2X) 


II.  X  (in  metres)  =  18400  log  ^  ML  + 
(1 +  0-00256  cos  2  X) 


X 


188  APPENDIX. 


THE  ASCENSIONAL  FORCE  OF  A  BALLOON. 

Lot  V  denote  space  occupied  by  the  gas,  v  volume  of  the 
solid  parts  of  the  balloon,  w  weight  of  the  same,  w1  weight 
of  the  aeronauts,  a  and  a)  weights  of  unit  volumes  of  air  and 
the  gas  respectively  at  0°  C.  and  76  centimetres  pressure, 
h  air-pressure  at  time  of  ascension,  hf  air-pressure  at  the  alti- 
tude where  the  balloon  is  in  equilibrium,  P  ascensional  force  ; 
then,  changes  of  temperature  being  neglected,  — 


II.  (jr+v)  =  r          +  „  +  „,. 

This  last  formula,  taken  in  connection  with  the  barometric 
formulae  before  given,  enables  us  to  find  roughly  the  size  of 
the  balloon  necessary  for  reaching  a  given  height  with  a  given 
load.  It  is,  however,  of  but  small  practical  value.  In  the 
upper  regions  of  the  air,  the  pressure  of  the  gas  in  the  balloon 
may  become  so  much  greater  than  that  of  the  external  air,  that 
the  aeronaut  will  find  it  prudent  to  allow  some  of  the  gas  to 
escape.  Moreover,  aeronauts  are  often  obliged  to  throw  out 
ballast  under  circumstances  which  cannot  be  foreseen.  The 
changes  of  temperature,  also,  in  different  air-  strata,  which  are 
very  considerable,  are  not  taken  into  account  in  the  formula, 
and  cannot  be  allowed  for,  being  largely  determined  by  the 
winds  and  atmospheric  currents  which  happen  to  prevail 
at  the  time. 


THE   END. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  5O  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


1946 


' 


SEE  27  I93b 


GUI  xi  ii 

M 

• 

V 

170ct'56GB 

DEC  27  i 

OC-f1*'-^    I    r-k 

OCT    a  1341M. 

V  V  *          **^ 

JUL  °4   t042 

r 

...    n        Af\  Jf% 

APR    23  1943 

ifcil 

VA  01160 


865704 


s* 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


